To Tailgate or not to tailgate.

If the truck camper is designed to run on a short bed it doesn't need the support so it is just extra weight. If you leave the tailgate on you are running the risk of putting a load on it that it is not designed to take. The tailgate is obviously designed to take some loading while down, but not much as it supported at the rearmost edge by crimped cables. I'd check the truck manufacturer load limits for the tailgate in the down position, where it could get overloaded. Me I'd take it off and store it somewhere safe and secure.