The ID of the flag with "In God We Trust" on it, which I have seen personally, is tenuous. One of the stars has a "July 4, 1863" date on it which could mean the Battle of Helena or a Vicksburg surrender and Iowa troops were at both places.
Several flags were taken at Helena; one of them by a Missouri regiment who sent it to the Mercantile Exchange in St. Louis (it has not been seen since) and I forget what unit that was offhand (I do have a file on it however). This includes the blue flag with red St. George's cross flag which we do not consider a Polk variant but, rather, a Trans-Mississippi variant of the so-called Taylor flag (probably more likely the flag of Walker's Texas Division). Another of this pattern exists for Dobbins' 1st Arkansas Cavalry.
I have not thought about these flags for some time but have files for all of them.