One other point based on the original post. An LLC is not secret. I can go to the AZ Corp Commission web site and see all the members. A trust, can have a one page "declaration of Trust" and that is all that is needed to show anyone. As far as the ATF, as M4 mentioned if you have one responsible party the paperwork is the same as am individual except you include a copy of the trust.
There is one more game people are playing and it is perfectly legal. Create a master trust with your spouse, 4 children, all your aunt sand uncles as trustees. make a carbon copy of that trust with you as the only trustee and the beneficiary is the master trust. Name each "sub-trust" with something like NFAHybridxxxxxx where xxxxxxx is the serial number.
The benefit is you are the only responsible party so only one set of finger prints. when you die, all your NFA items are left to the master trust named "MrSmithScrewsTheMan" and all the trustees named have access to all the NFA items.
Think about your needs and your future but don't do an LLC, way to public go trust instead.