I do not find a Lee associated with Rivers' Battery. At the time of forming Lee's Battery, the 1st Arkansas was out East having fought at Stones River December 1862. Here's some rather disjointed information. Sorry,

George Martin

First Arkansas Light Artillery (Rivers’ Battery)

"The information on these pages was researched and edited and graciously given to the Edward G. Gerdes Arkansas Civil War page by Bryan R. Howerton, who we thank so much. He can be reached and thanked at this email address! If you have questions regarding this unit, please talk to Bryan. "Bryan R. Howerton"

The First Arkansas Light Artillery was organized at Fort Smith, Arkansas, on September 14, 1861, but continued to recruit and enlist members well into October. The original officers were Captain David Provence, First Lieutenant Stephen D. McDonald, Second Lieutenant John T. Humphreys, and Third Lieutenant William H. Gore. Most of the officers and men had recently been mustered out of the Arkansas State Troops. The battery fought at the battle of Pea Ridge (Elkhorn Tavern), Arkansas, in March 1862, before being ordered east of the Mississippi River, where they spent the remainder of the war.

On November 18, 1862, the battery received a large number of transfers and details from the Arkansas regiments and battalions of McNair’s Brigade. Many of these men are later listed on the battery’s rolls as deserters; however, many, if not most of them simply returned to their original regiments.

In keeping with the practice in the Confederate army of referring to an artillery battery by the name of its current commander, the First Arkansas Light Artillery was variously known as Provence’s Battery, Humphreys’ Battery and Rivers’ Battery. The Compiled Service Records of the men are filed under “Rivers’ Battery,” Microfilm Roll #41.

Captain David Provence resigned to accept an appointment as colonel, 16th Arkansas Infantry Regiment. His resignation was formally approved on June 10, 1862, at which time Lieutenant John T. Humphreys was promoted to captain. Captain Humphreys resigned on October 15, 1863. Lieutenant John W. Rivers, who had risen from the ranks, was promoted captain, effective October 15, 1863, and commanded the battery to the end of the war" [http://www.couchgenweb.com/civilwar/1starty.html]
......................

Rivers' Battery, Arkansas Light Artillery

Provence's-Humphrey's-Rivers' Battery was organized in the summer of 1861 with men from Fort Smith, Arkansas. The unit was engaged at Elkhorn Tavern, then moved east of the Mississippi River. After being involved in the Kentucky Campaign, it was assigned to McNairs' Brigade and later J. W. Eldridge's Battalion of Artillery, Army of Tennessee. It saw action at Murfreesboro, Chickamauga, and Chattanooga. In December, 1863, the company was ordered to Atlanta where its guns and equipment were redistributed and the men assigned to other artillery commands. Its captains were John T. Humphrey, David Provence, and John W. Rivers. [http://www.itd.nps.gov/cwss/soldiers.cfm]

..................

Humphreys, John T—Second Lieutenant; enlisted at Fort Smith, Arkansas, September 14, 1861; promoted first lieutenant, April 10, 1862; promoted captain, July 13, 1862; resigned, October 15, 1863; died in California.

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

Roswell W. Lee graduated from West Point with the class of 1833 and appointed a 2nd Lieutenant in the 3rd Artillery taking part in the Florida war, he resigned his commission in 1838. He settled in Texas some 20 years or so before the CW. [I haven't been able today to download his Academy records, will try again tomorrow]

Roswell W. Lee, AAAG, Colonel Douglas H. Cooper's command was wounded slightly in the battle of Chuto-Talasah, on Bird Creek, Indian Territory, December 9, 1861, later reported as Captain Lee's Light Battery on the staff, 1st Indian Brigade, Indian Division, District Indian Territory, n.d., on May 29, 1863, his battery, "not fully organized," was stationed at Elk Creek in Brigadier General William Steele's command, March 16, 1864 reported as Inspector General, Gano's Brigade, District of Indian Territory, IG First Division (Indian Troops) District of Indian Territory March 16, 1864, later reported, n.d. as Colonel R. W. Lee's Command on N. W. Frontier of Indian Territory
[National Archives M331 Compiled Service Records of Confederate General and Staff Officers, and Non-Regimental Enlisted Men, Roll 155, his records contain a number of handwritten letters, a couple being his asking for appointment as Colonel PACS, too faint for me to read on-line]

O.R.-- SERIES I--VOLUME XXII/1 [S# 32]
DECEMBER 7, 1862.--Battle of Prairie Grove, Fayetteville, or Illinois Creek, Ark.
No. 36.--Reports of Maj. Gen. Thomas C. Hindman, C. S. Army, commanding First Corps, Trans-Mississippi Army, including preliminary skirmishes.

. . . I had with me the following staff: Col. R. C. Newton, chief of staff; Maj. J.P. Wilson, assistant adjutant-general; Lieut. S. B. Reardon, aide-de-camp; Lieut. R. W. Lee, aide-de-camp, acting chief of ordnance; Col. D. Provence, acting chief of artillery

..........

Lee's TX Light Artillery Battery was reported in Cooper's Brigade, Steel's Division, District of Arkansas from April 1863 to October 1863

Troops in Steele's division, April 30, 1863
Cooper''s Brigade.

Brig. Gen. D. H. COOPER

1st Cherokee, Col. Stand Watie.
2d Cherokee, Col. W. P. Adair.
1st Choctaw and Chickasaw, Col. Tandy Walker.
1st Creek, Col. D. N. Mcintosh.
2d Creek, Col. Chilly Mcintosh.

Cooper's Brigade--Continued.

1st Chickasaw Battalion, Lieut. Col. L. M. Reynolds.
Osage Battalion, Maj. Broke Arm.
Seminole Battalion, Lieut. Col. John Jumper.
5th Texas Partisan Rangers, Col. L. M. Martin.
29th Texas Cavalry, Col. Charles De Morse.
Gillett's cavalry company, Capt. L. E. Gillett.
Scanland's squadron, Capt. John Scanland.
Howell's (Texas) battery, Capt. Sylvanus Howell.
Lee's light battery, Capt. R. W. Lee.

Oct '63 to Jun '64, reported in Cooper's Brigade District of Indian Territory

HEADQUARTERS DISTRICT INDIAN TERRITORY,
Fort Towson, January 15, 1864.

Lieut. Gen. E. KIRBY SMITH,
Commanding Trans-Mississippi Department:

........I need a first-rate inspector; General Steele left none. Capt. R. W. Lee (a graduate), a man of fine sense and great energy, has agreed to serve. His habits were once occasionally bad; for twelve months they have been good, and I have no doubt will continue so. I hope you will give him as much rank as you can with the understanding above. He is under obligations of a private nature to me that will keep him true to his word. (So much as refers to him I hope you will consider confidential.) He ought to have at least two good assistants. No part of your department needs efficient inspectors so much as this.

Very respectfully, your obedient friend and servant,

S. B. MAXEY,
Brigadier-General, Commanding.
OR V34Pt2p. 877

HEADQUARTERS DISTRICT INDIAN TERRITORY,
Doaksville, Choctaw Nation, January 16, 1864.
Capt. R. W. LEE,
Insp. Gen. and Mustering Officer, 3d Choctaw Regt., Present
OR V34Pt2p. 882

R. W. LEE,
Assistant Inspector-General, District Indian Territory.
February 26, 1864

.............

D. PROVENCE,
Provost. Marshal. General, District of Arkansa
February 16, 1865.

.............

Stone's River after battle report:
Report of Capt. John T. Humphreys, Arkansas battery.

CAMP NEAR SHELBYVILLE, TENN., January 10, 1863.

SIR: In obedience to orders, at daylight on the morning of December 31
[1862,] my battery advanced to the position occupied the day previous
by Robertson's battery. Your brigade, then advancing, engaged the
enemy and was rapidly driving him back. I followed as closely as the
nature of the ground would admit, but, turning to the right too suddenly,
became separated from your brigade, and, having advanced in a
northern direction about 1 mile, engaged a battery of the enemy, a part
of which was soon after captured by our advancing lines. I then passed
through an interval in our infantry lines caused by our left swinging too
far in that direction, and, taking position near Mr. Cowan's house, some
3 miles form and northwest of our original lines, opened fire upon the
enemy's batteries, some 500 yards distant, in position on an elevation
equal to our own and partially concealed by a narrow skirt of timber
intervening between our position and his. The enemy's guns (supposed
to have been sixteen in number), then firing upon our infantry and other
troops in his front, were immediately turned upon us with great
precision and rapidity. We responded as fast as our guns could be
served, and for more than half an hour drew the entire fire of all the
guns on the opposite hill. Two of my guns were rendered useless by
ammunition too large, and two others were, during the action, disabled
by the enemy's shot, one having a wheel shot off and otherwise injured,
and the cannoneers of the other being in the same way disabled. Five
horses were killed by an exploding shell. With two pieces (a 3-inch rifle
and 6-pounder gun) we maintained the fight until our advancing lines
were charging the enemy's guns, when we were ordered to fall back,
which we did, to a point some 300 yards in rear of that position. Eight
men were disabled by wounds.

During the engagement there were many acts of individual gallantry
displayed, some of which I beg leave to mention. Lieut. John W. Rivers,
when the cannoneers at a piece were disabled, seized the sponge-staff,
and, calling other to his aid, filled the post of No. 1 with energy and
determination.

First Lieut. William H. Gore was acting as gunner when he was knocked
down and wounded by a shell. He immediately rallied and called upon
the men, wounded like himself, to rally to their work.

Lieut. Oliver P. Richardson, whose gallantry on the field of Richmond,
Ky., attracted the attention of Gen. Churchill, commanded his section
with ability.

Lieut. Henry C. Riggin, after Lieut. Gore was wounded, was
instructed with the command of that officer's section, and rendered
efficient service through the day.

All the above-named officers in this engagement behaved with a
gallantry and displayed a devotion to cause and country which should
entitle them to the favorable consideration of the commanding general.

First Sergt. William Shea was fearless and efficient. Ensign Cameron,
when a part of the flag-staff was shot down, observed coolly that they
were "shooting a little close."

Private Samuel M. Tucker had a sponge-staff shot out of his hand, and
by the same force he was knocked over, but immediately returned to
duty. Private Joseph W. Adams was wounded in the foot and had his
pantaloons torn off by an exploding shell. Dennis Corcoran was severely
wounded in the neck by splinters from a shell, and reluctantly was
compelled to leave the field.

Duty Sergeants Thompson, Casey, Greer, Long, Brewer, and Burkett
are all deserving of favorable mention for high courage and efficiency.
In fact, to do ample and full justice I should have to mention every
name on the company rolls. All were in trying positions, and suffice it
to say that no men could have better while subjected to a terrific fire of
shot and shell which tore down threes like the whirlwind and scattered
them like the lighting.

Fortunately while horses and guns were destroyed the men were
protected by the "Giver of all victories." It is to that power and the
officers and men mentioned referred to that the success of this battery
can be ascribed.

Very respectfully,

JOHN T. HUMPHREYS,
Capt. Artillery.

Gen. E. McNAIR,
Comdg. Third Brigade, McCown's Division, Army of Tennessee.

Source: Official Records
CHAP. XXXII.] THE STONE'S RIVER CAMPAIGN. PAGE 957-29
[Series I. Vol. 20. Part I, Reports. Serial No. 29.]

Chickamagua after battle report:

Report of Capt. John T. Humphreys, First Arkansas Battery.

CAMP NEAR CHATTANOOGA, TENN.,
October 7, 1863.

SIR: In obedience to orders received this morning, requiring a
report of the part taken by my battery in the battle of
Chickamauga, I have the honor to report that on Saturday,
September 19, I was ordered about 3 p. m. to follow my brigade
(Clayton's) and to keep up with it. The brigade moved forward
rapidly through a piece of thickly wooded land in the direction
of the Chattanooga road. Leaving the caissons in rear, I follow
immediately in rear of the brigade, which, when it had advanced
about 300 yards, was checked in its progress by the enemy in
front, with whom it was hotly engaged with small-arms. I formed
the battery first in line, then in battery immediately in rear of the
brigade, but was unable
to fire because our men were precisely between the guns
and the enemy. The enemy then gave way, and First Lieut. John
W. Rivers, of my battery, was ordered by Gen. Clayton to carry
a message to Gen. Stewart to the effect that the enemy were
being driven, and that he (Gen. Clayton) wanted more men.
Lieut. Rivers, supposing that I had been notified of the order to
him, left to obey it. The brigade of Gen. Clayton soon advanced.
With one officer (Lieut. Richardson), I followed it closely about
a quarter or half mile to the Chattanooga road, where I
immediately caused the guns to be unlimbered and put in
position. Our own men were much scattered, their ranks much
broken; some were a short distance in front, some on the right
of the battery. At this point, the enemy opened upon us a hot and
well-directed fire, which disabled two limbers--one of a
12-pounder howitzer, another not fire without dangerous
consequences to our men, who were scattered in groups along
our front, and in small parties were coming back from the
charge.

Finding that our men in disorder were retiring upon my right and
leaving me without support, I caused the guns to be limbered up.
or such of them as there were horses left to move, and taken to
the rear. I then sent to the rear for caisson limbers to take out the
two remaining pieces, and to urge them forward rapidly, sent
Lieut. Richardson, my only remaining lieutenant, who stated at
the time that he disliked to leave me alone in such a perilous
condition. I then rode some 150 yards to the right and rear of the
pieces left in the road to where Col. Woodruff, of the
Thirty-sixth Regt. Alabama troops, was endeavoring to collect
his scattered command. I asked him for men to aid me in hauling
off the pieces, then about equidistant between the enemy's lines
and our own. He said to me he would, but, "see, these are all
the men I have," pointing to a few whom that gallant officer
was assembling around him. I then went back to the pieces and
found that one limber had come up and saw it take off one of the
guns. But one piece now remained in peril. I then went back and
about 300 yards to the right of my gun, and 100 or 200 to its
rear I met Gen. Hood; told him that the gun would be lost if it
remained much longer where it was. Gen. Hood asked me who
ordered me into that place. I told him. His responses was, "You
had no business there." I told him I knew that, but asked that he
would press forward his lines sufficiently far to cover the gun.
He said, "If it is lost we will get it back again." I then started
back to the gun, and met it about 100 yards from where I had
left it being carried out by a small number of troops of Hood's
division and a few of my men, one of whom (Private William N.
Green) I have had occasion on other battle-fields to mention for
distinguished services.

I lost 1 man killed, 2 wounded slightly; some half dozen were
injured besides, and though disabled at the time, have since
returned to duty, and are not mentioned in the list of casualties.

I was not notified of the order depriving me of the services of
Lieut. Rivers. That officer, so soon as he had conveyed the
message above named, rejoined the battery.

I could not and did not during the day fire a single gun, though
several times urged to do so. It is a subject, situated as I was, of
self granulation that no Confederate was injured by my guns,
which would have been the case had I fired at any time during
the day.

I append a list of the number of men taken into the field
on Saturday: Commissioned officer, 3; non-commissioned
officers and privates, 86.

On Sunday the battery was not engaged and exposed to no fire
of small-arms.

Lies of men and officers taken on the field on Sunday:
Commissioned officers, 3; non-commissioned and privates, 79.

Very respectfully,

JOHN T. HUMPHREYS,
Capt. First Arkansas Battery.

Maj. J. W. ELDRIDGE,
Chief of Arty. of Gen. Stewart's Div., Army of Tenn.,

Source: Official Records
PAGE 411-51 KY., SW. VA., TENN., MISS., ALA., AND N. GA. [CHAP. XLII.
[Series I. Vol. 30. Part II, Reports. Serial No. 51.]