Really, its more likely that a local or state le will ask for paperwork than an atf agent. They dont just go around looking for nfa weapons to see if they are registered or not.
So, while not necessarily illegal to not have the paperwork, it is what proves you not guilty at the moment of le interaction.
This is also going to depend on the officer. Finding an sbr in a car full of gangbangers isnt the same as being two lanes down at the local range from joe citizen with an sbr. Joe citzen, who isnt concerned with who sees it in public, and still isnt concerned if an le officer walks up and identifies himself is either legal, or woefully ignorant.
Also, you can carry a copy, and nothing says it has to be full sized. I shrink copies by 50%, then cover it with plastic, either laminating, or "poor boy" laminating with clear shipping tape, to make it last longer. The smaller sheet is much easier to take along. A friend shrinks his until it fits in his wallet but i can hardly read it. Personally, i like to stick with easily readable.
If you have a trust, you should also carry a copy of the page that lists the trustees. The f4 only proves the trust owns it, not that you are a qualified person to possess it. The trustee page establishes that. In all cases, your id shows that you are the named person.