The bottom line:
1. You are not breaking the law by not keeping keep the NFA paperwork with the NFA item.
2. It's a very good idea to keep the NFA paperwork with the NFA item to avoid problems
3. The range officer was justified in kicking someone off the range who doesn't have the proper NFA paperwork. Private organization, private rules - it could require more than a minimal compliance with NFA laws, i.e. you must have the NFA paperwork to bring NFA items to the range.