For temporary export to the U.S. for the purposes of hunting or competition, you can Form 6 it yourself. Then you need an ATT to the Border and back, and you’re good to go. There is a time limit on that; don’t recall if it’s 90 days or 6 months max.
As others have said, there is a prohibition on permanent transfer in the ITARS and another piece of legislation. One refers to them as “weapon of war”, another calls them an “offensive weapon for the purpose of combat” or something equally obtuse. Either way, I don’t know that you can take any part of it down permanently. Stripping the parts out of the lower and bringing them down still runs into conflict with ITARS, I believe.
The fact that you’re immigrating may give you a bit more latitude depending on what status you’re coming in on (i.e. resident alien, landed immigrant, etc)