Ben,
The flag of the 44th Miss. taken is War Department 209 and is one of the Second Nationals made in the Winter of 1864/1864 most likely that were typical of a number of Mississippi units in the Army of Tennessee/Army of Mississippi (later Polk's Corps).
This flag can be found in the collections of the state museum of Mississippi.
I also have the 12th Iowa taking the flag of the 1st Louisiana Battery but am not sure how sound that ID is.
Greg Biggs