Allan,

This is one of those obscure units that frustrated me for a long time.

I was finally able to piece together a rough timeline from its formation to consolidation with Hawthorne's regiment as follows:

From:
University of Arkansas special collections-civil war manuscripts
Catalogue for the Manuscript Collections for the Civil War
39. Confederate States Army. Staff Officers. Selected compiled records, 1861-1865; 2 rolls.

[paraphrased extract]
In a letter from Little Rock, dated Jan. 7, 1863, to his old commander Gen. Hardee, Col A.T. Hawthorn continued his campaign for promotion to Brig. General....giving a summary of events since leaving Hardee and returning to Arkansas.
He states that he was appointed Major of an unarmed Battalion, and then two weeks later elected Colonel of a Regiment whose Colonel had resigned. [Johnson's regiment]

The unarmed battalion referred to by Hawthorne would become Chew's battalion.

From:
The Alexander Papers,
paraphrased extracts]

this battalion can be traced as follows:

Oct 19, Camp Reagan
"...............................................
Colonel A. T. Hawthorne is assigned to the command of the battalion composed of the companies of Dawson, Tumlinson, Perkins and Hay...est ?[could not decipher] of ...................
..............Hawthorn's battalion Arkansas Volunteers..............."

Oct 23, 1862
To Colonel C. W. Adams cmdg. 3rd regt Ark Infantry thru Colonel Brooks-
This message countermands the order assigning Captain Chew as LtColonel of Adams' regiment.... "in deference to the wishes of men and officers of the regiment in their memorandum to the Major General commanding."

Nov 2
To Colonel Dobbin from Gen Hindman
Informs Dobbin that Captain Chew will be sent to assist him in the distribution of arms and ammunition [enroute from Little Rock] to the unarmed men of Brooks' brigade, which includes Hawthorne's battalion.

Nov 3
From camp on Mulberry to Col Brooks' camp of unarmed Infantry-
"...... Assign the senior Captain to the command of the Battalion now commanded by Col Hawthorne and move them up when armed. You will notify the Captain commanding that the battalion will be consolidated with Buster's battalion and a regiment formed of the two. The field officers will be appointed hereafter."

Nov 5
From Brooks' hq at Camp Hindman
Memorial of commissioned officers of Hawthorne's battalion to have Captain Chew appointed Major to fill necessary... by promotion of Major Hawthorne.

Dec 1
Fagan asks for the consolidation of Major Chew's Arkansas Infantry battalion and Hawthorne's Arkansas Infantry regiment both of his brigade .............................

Dec 14
Hindman's endorsement regarding this consolidation:
"Chew's Battalion consists of four companies and had been selected as a Battalion of sharpshooters. I assigned Major Chew to the command of it sometime ago. In the late battle of Prairie Grove the Battalion acted with and as a part of Hawthorne's regiment, and Major Chew was killed.....Hawthorne's Regiment is much reduced and the Battalion if attached to any Regiment would probably prefer this one due to prior service together. earnestly recommended-TCH "

Dec 23
Hindman to Gen Holmes-
"In Fagan’s command there is a battalion of five companies formerly commanded by Major Chew, which I organized as a Battalion of sharp shooters. It acted with and as a part of Hawthorns Regiment in the recent battle and Chew was killed, and as Hawthorne Regiment is much reduced and the men of Chew's Battalion would prefer to go to that Regiment, I ask that the Battalion and Regiment be consolidated."