Re: Carry paperwork with NFA weapon??
There is no FEDERAL requirement that you carry the registration document (Form 4 or equivalent) with you when you take the gun off your property, only that it be made available to an ATF agent on request. BUT, having that paper might protect you from hassle by local police. I say "might" because some police are totally ignorant of firearms law and believe that all machineguns are illegal, or should be, and will arrest anyone with such a weapon.
I will go one step further than Alpo, in that I always kept my registration forms in my safe deposit box, and carried copies with me. I never had any problem, but then I never was stopped, either.
You don't need that paper to "prove" the gun is registered; the entry in the NFRTR does that. So even if you lose the paper, your gun is still registered, and you can get proof of the registration from BATFE at any time. Your approved transfer/registration paper is simply your copy.
As to your local range policy, that is set by the club or organization operating the range.