You asked, "Was the A.o.T.'s need for troops any greater than those of the A.N.V.?" Lee took twelve divisions of infantry into Pennsylvania. After sending Stevenson's Division (the largest in his army) to Mississippi in December 1862, Bragg was left with five divisions. You said, "The same with Bragg not wanting to send troops from his Army in Tennessee to Joe Johnston in Mississippi in 1863." Bragg sent Breckinridge with another division to Mississippi in May. That left his AOT with four divisions to contend against Rosecrans with 92,000, unable to contend against a flanking attack in Middle Tennessee, or an advance by Burnside from Kentucky into the Cumberland Plauteau. Burnside's IX Corps began moving from North Carolina and Virginia into Kentucky during the last week of March 1863.

So who really needed these troops in May 1863? Eventually a decision had to be made in favor of the AOT after Bragg retreated over the Tennessee and Chattanooga had to be given up. Four divisions just weren't enough to contend against four corps of the Army of the Cumberland. Nearly all of the troops assembled under Johnston went to north Georgia, along with two divisions of Longstreet's Corps. Even with all its accessions from Virginia, Mississippi, East Tennessee and the coast of South Carolina, that combined army did not surpass the one Lee led at Gettysburg.

What might have happened if Johnston had followed instructions to take command of the AOT in Bragg's absence? In early April 1863 Johnston happened to be ill, but troops from Loring's Division were in transit from Jackson to Tullahoma at that time. They were immediately returned at the appearance of a threat to Grand Gulf in late April. What if Stevenson's Division had been returned when the Federal attempt to force a passage along the Sunflower and Yazoo Rivers failed, and all those brigades later sent to Johnston in Mississippi had gone to East Tennessee instead?

In this scenario, the AOT would have included three corps of roughly three divisions each. Johnston could have left Bragg in Middle Tennessee with four divisions, and moved troops through the Sequatchie Valley and Cumberland Gap. Five divisions plus strong cavalry support from Morgan and Wheeler might have been enough to overwhelm individual divisions under Burnside then advancing towards the Cumberland Plateau. A series of Federal defeats on Kentucky soil would have forestalled any plans for an advance into middle Tennessee, and probably obliged troops intended for Grant's army to be sent to reclaim that state. Plans for the impending campaign on Vicksburg might have been put off. It's even possible that troops could have been detached from the Army of the Potomac, as they were after the disaster at Chickamauga.

As reorganized after Chancellorsville, the ANV should have been sufficiently strong to carry on offensive operations in Pennsylvania. No need to detach Longstreet's Corps. Johnston wasn't an ideal leader for an offensive, but a campaign in Kentucky would have preempted plans made in Washington for military operations in Tennessee and Mississippi. It's stretching things a bit, but in hindsight consider -- all the troops sent to Mississippi in May 1863 went there for no useful purpose, and many of the troops captured at Vicksburg and Port Hudson were lost to the Confederacy forever. And what if Morgan had gone raiding in Kentucky with a real army nearby to support his efforts?

Anyway, it's entertaining to consider the possibilities.