Regardless if it is required to carry a copy of your stamps, you know it can't hurt anything. Its no skin off your teeth, just carry them in some fashion and it could save your some headache. When my stuff all gets approved, I will carry a copy in my bags.
Side story: A few years back I was shooting at the NRA head quarters range in Northern Virginia. Guy in the next lane was shooting suppressed hand guns. Mid way through my session, one of the RSO's comes over to the guy and asks him if you had his paperwork for his supressors, which he said he did. Then the RSO said he should have informed them. The guys said sorry, and that was it. RSO went away. I never noticed any signage about that, nor remember signing anything that said to declare NFA. Its the NRA building for Christ sake. Are they verifying every gone is not stolen and no one has a criminal record too? Moral of the story, people can be D-bags anywhere. Carry your paperwork.