While researching capture papers several years ago we found some regulations regarding sending back full auto items. If I remember correctly, anyone could bring or send back full auto items until some point in 1944 where the Army realized this was a problem. After this circular was distributed, only officers could technically bring or and back full auto. The Army was most concerned with the sending back of concealable pistols, that’s why we don’t see much paperwork on rifles. I think I have this all saved on an old laptop, I’ll look for it.
I have seen papered full autos. I have heard rumors you could register an MG to this day with the capture paperwork. Not certain on that.