You do NOT have to have a copy with you, but it is a good idea. You do NOT have to have the original, and in fact there's NO WAY I would carry the original around with me. I put photocopies of all my forms in my range bag and take all of them with me whenever I go out to shoot anything. That way I know I have all of them with me all the time.
All it takes for any LEO to check them is a quick call to the local BATF office and they can quickly confirm that in fact the guns are registered to you. Remember, those documents are really TAX FORMS and as such contain privileged personal information that can only be disclosed to specific parties. I don't think a local LEO can call BATF and get the information directly but I'm not sure of that.