You're probably good to go, but, there's more than just comparing trailer weights to the truck's advertised capacities.
Those capacities were for the truck as it left the factory, and do not allow for weight of anything but a driver. Additional people, aftermarket accessories, cargo, and a weight distributing hitch, all reduce those capacities, pound for pound.
Installing lift kits and changing tire sizes, have an impact on those numbers.
If the truck has not been modified and you don't have a lot of added weight (passengers, pets, cargo, etc), you should be fine. Your weight estimate for camper load, is probably pretty close.
To get your actual payload available, take it to a scale, get it's actual weight and subtract that from it's GVWR. What's left is available for people, pets, cargo, hitch, and trailer tongue weight.
With weight distributing hitch, your trailer will need close to 800 lbs out of your available payload.