If you want to keep it nice and legal and retain ownership in the long run, what you're supposed to do is turn it in to the ATF and request an exemption for it not having been registered when it should have. The ATF retains the weapon while they make their decision. You will have to explain where it's been and why it wasn't registered originally.
If they decide in your favor, you fill out the paperwork, pay the $200, and it's yours.
If they decide against your, they keep it. They won't destroy something like an STG-44. They will retain it.
A woman I work with had a similar incident. Her Grandpa, a WWII US Army vet, died and amongst her belongings she found an odd Mauser pistol. I ID'ed it as an "M1932" Schnellfeuer.......a select-fire Mauser C96 chambered in 7.63x25mm. Came with the cleaning kit and wood n' leather shoulder holster that doubles as a detachable stock. She turned it into the ATF with an explanation of the circumstances. Ultimately, it was registered and returned to her, so I got to cross a gun off my bucket list of "Shit to Shoot Before I Die".