I've been looking a little more into this and have noticed that it changes a bit from state to state.
For example, in Texas most NFA stuff is specifically illegal unless they are federally registered (via F1 or F4). So if you can't produce your forms in TX, it's game over.
Knowing that, I looked up FL law (I live in FL) and it's pretty vague. SBR's, SBS's, MG's are specifically defined by law- but suppressors are not even mentioned.
Even the ATF site suggests carrying a copy of forms with you. I'd imagine that if they suggest carrying a copy, it's not necessary to carry originals- so that clears that one up pretty well.