I don't know how anyone can say something like my dog would never hurt anyone. It may be true that your dog has never yet hurt anyone. But we can't even say that about people. People are supposed to be controlled by their brain and rational thinking, unlike animals who are supposedly controlled by instinct. Yet people who have never been violent, sometimes go crazy and do really crazy things. If that can happen to people, it can surely happen to animals.
That is why animals are supposed to be kept under control (on a leash) at all federal properties and most state properties. And yes, I own dogs, and yes I've sometimes taken them camping. Follow the rules. The rules were meant to protect all of us. If the rule is keep it on a leash, then do it. If the rule is your breed of dog isn't allowed, then don't bring it. If you have a service dog, then be very clear about that fact, and have the paperwork to back it up. It is really quite simple.