North Korean, East German and "North Vietnamese" SKSs are all understood to have arrived in the US exclusively via bringbacks from Vietnam, so no paperwork required, but it's a nice bonus.
There aren't any other country's SKSs that are as clear cut as the Trinity rifles. No import marks, patina, battle damage all indicate to very strongly suggest that they were bringbacks, but one can't be 100% certain with any of them. Generally, bringbacks are prefaced by the adjective "likely" in those cases.
I suspect these DD-603 papers would be in personnel service records if they exist at all. Alternatively, they could be in the accujmulated unit records for those units that handled the approval of the export of the war trophy firearm. In either case, yes it would probably take a FOIA request including fees and labor. That could stack up for difficult to locate records...if they are found at all. Probably not worth trying just to satisfy curiosity or even to try and add value. Best you could get is a photocopy which wouldn't add much value in any case.
My $0.02