The rectangular AOT flags were from units of Brown's Division who began their attack directly to Cleburne's left and became a bit intermingled as they hit the final line after overrunning the Union advance line held by part of Wagner's Division. Hence the two patterns in the painting.
The flag you mean should be that of the 5th/13th Arkansas which was captured at Franklin by the 88th Illinois. The soldier who took the flag held it as late as 1888. Now, owned by someone else, it was displayed at a Civil War show some years ago and Lee White, a historian at Chickamauga/Chattanooga National Military Park saw it and gave me a drawing of it for my files. It is one of the March, 1864 issues with the rectilinear moon bearing the crossed cannons battle honors (not inverted) and the designation 5th & 13th Ark Regt's. The battle honors are, from top to bottom - Belmont, Shiloh, Perryville, Murfreesboro, Richmond, KY., Liberty Gap, Chickamauga, Tunnel Hill, Tenn. and Ringgold Gap.
Sure wish I had its measurements and a picture of the actual flag. Its flag staff also survives but it is held by someone other than the flag's current owner. That I have seen.
You are correct that Cleburne's Division got new Hardee flags in September 1864 prior to the Tennessee Campaign. These reverted to the circular moons and at least some bore crossed cannon honors (16th Alabama) while several were unmarked (based on the surviving flag of the 33rd Alabama and two unknown Franklin captures at the MOC).
What had been Mercer's GA Brigade, Smith's in later 1864, which was joined to Cleburne's Division late July 1864 after the breakup of WHT Walker's Division following its casualties in the Battle of Atlanta, were also issued these new Hardee flags but they did have unit designations on them, at least based on the surviving flag of the 54th Georgia Infantry. Prior to this, the brigade had been carrying ANV style flags as made by the Charleston Depot.
Hope this helps.