The Van Dorn battle flags had been replaced by other patterns by 1864. What Cockrell's Brigade carried in the Tennessee Campaign were the 12 star Mobile Depot battle flags that were rectangular. The brigade was serving in the Dept. of Alabama, Mississippi and East Louisiana before coming over to Georgia as part of the 4 divisions sent to Joe Johnston (Polk's Army of Mississippi's two infantry divisions, one division from the Mobile Garrison and William "Red" Jackson's Cavalry Division). All of the units carried these flags.
Please see our CS flags website at www.confederateflags.org for what these looked like.
As to the flag of the 2nd/6th Missouri it was taken at Franklin by Sgt. Alfred Ransbotton of the 97th Ohio Volunteers. The 88th Illinois captured the flag of the 1st/4th Missouri and it was sent to Chicago where it burned up on the post-war Great Chicago Fire. It too was a Mobile Depot pattern.
The flags taken at Franklin, Nashville and Ft. Blakely are still pretty confusing on several levels.