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You cannot get permission to take a Swiss arms into the U.S. Under any circumstances. Us state dept will not allow them in at all. Worth a try ( applying for permit)if you like just know your chances of succeeding are next to zero
If memory serves me correct, to make the Swiss Arms 922r compliant, you would have to switch out 7 parts for U.S. made ones (as it has 17 listed foreign made parts, and you can have no more than 10 for a compliant rifle). This could be accomplished by using a U.S. made magazine (3 parts), Sig Sauer 556 lower (1 part), butt-stock (1 part) and pistol grip (1 part), then you just need to find one more part to switch out. I'd suggest basing it off a Swiss Arms carbine, and have a barrel made from a U.S. made blank (not 100% sure if this would count, but it might) at 18.5", including an integrated muzzle break. I've also been advised that the BATF can be fussy about letting rifles with bayo lugs an tritium sights in, so depending on the model, you may have to remove these features. I'm not 100% on all of this though, as I've only had a few conversations with some American friends on the subject.
Are you talking importing (moving) or a Form 6 NIA?
If you are looking at taking a swiss arms black special for a competition on a form 6 NIA then it absolutely can be done - as my current form 6 has mine on it, and I've been to the US last year in December for a competition with it.
If you are talking moving to the US then as people have already stated you are SOL unfortunately.