You just register your SBR, then when you travel out of state with it, ATF likes to be notified. I'm not sure if you need a reply from them, I'd call and ask about that, they're pretty good and giving you the info you need. ATF.gov also has info, you should check that out too. But it's only a few minutes to find out where they want it mailed and if you need to wait on a reply. If you need a reply, they don't take as long as the stamps do, thank god.
Since your case is special, perhaps they'd be okay with a general "I frequently travel back and forth with my SBR". You need to call and check. They may send you a reply saying they know about it all is good. Who knows? Unless you call, you won't.
FWIW, I knew of a guy that had a class3 business on the state line of FL and AL. He rented MG's and sold those and suppressors among other things IIRC. I installed his security. Wanna say his property literally strode the state line and he joked you shot in one state but the target was in another, but whatever the case, I'm sure he regularly crossed state lines and I'm sure many of his customers down there did too. In fact, I know he did, he had business licenses for both states (not sure how his FFL worked though). Really nice guy, bought my first integral Mk2 from him but had to unfortunately stop the process because I moved.
And if you have MG's or DD's, pretty sure they want to know where you're going, when you'll be back, etc., and I think it's a permission deal, not just notification. They're more strict with certain items like those. But even then, they may make an exception. They must have with that dealer I'm talking about. They can basically do what they want. I have a semi M2HB. Technically an MG, not a rifle or DD, but it has an exception because the mfg. asked for it. I have to keep the papers it came with stating such.
I don't think suppressors require a notification or anything, but many folks send one just in case.
I'd be sure to contact ATF and ask them though to be on safe side. But it's no problem traveling with them if you know what to do and CYA first if necessary.
Love my SBR's. I may be moving sooner or later too, possibly to a state line, so despite how this thread goes, when my time comes, I'll be calling ATF myself and getting the info when I'm ready. I think everyone should. Unless you get one after another solid unanimous responses, you should call ATF. Or your NFA attorney if you have a trust (you DID use an NFA attorney for your trust, right?). That's just my experience and $.02.