Not certain, but I highly doubt it...
If possible (can it be done) why not turn the long gun into a semi-auto from a full auto and keep it. Would this make the legal to own if converted?
For many, many years, the ATF has operated under the premise that "once a machine gun, always a machine gun". The only exception was a gun that they declared legally "dewat".
So, as long as the receiver is intact and functional, no matter what parts are with it, it is a machine gun under the law.
Posessing the unregisterable, illegal gun long enough to "convert" it is a crime.
There is no legal way out. None. The instant you lay hands on the hypothetical full auto bringback AK, you are guilty of a crime. It CANNOT be made legal under existing law. The only option is immediate surrender to the Feds (or any other law agency), and hope they do not prosecute. They can legally prosecute you. And they can convict you. Usually, they will not prosecute, as your voluntary surrender (as soon as you know what it is) is taken as a show of good faith that you had no intent to break the law. Delay in surrendering the gun is usually taken as the opposite. And then, they are much more inclined to prosecute.
The 1986 Hughes Amendment closed the civilian Full auto registry. NO gun, no matter how rare or historically significant may be registered to a private individual after May 19th 1986. None.
There is a possibilty that a historically significant gun
might be allowed to be registered to an agency or organization that already posesses the correct license, but only if the ATF approved it. And they are under to compunction to do so.
There was a case not too long ago where a WWII bomber was recovered from a lake, with its machine guns onboard. The museum that was recovering the plane wanted the guns, too (they were non functional after years under water) and the ATF fought them on it. I don't remember how it finally turned out, but I think the ATF allowed the museum to have the guns, but only after they were officially "dewatted". This case was, I believe, one govt sponsored group (museum) arguing with another (ATF), so a compromised was allowed. If it had been a private group or individual, I think the ATF would not have been so ....flexible.
In short, the hypothetical "discovered" bring back AK is a fed felony, just by its mere existance. ANYONE who posesses it is guilty. Even a licensed NFA collector cannot legally posess the gun, or register it.
A licensed NFA dealer
might be able to register the gun, for future transfer ONLY to law enforcement agancies or the US govt (that registry is not closed), HOWEVER, posession of the AK, unregistered is still a crime, and no NFA dealer is likely to risk this Catch-22. I think the ATF would take a dim view of this, even if it was legally possible, and I'm not certain it is.