If you only we're planning on hauling the camper, your truck would be fine for most campers available as evidenced by the thousands of single rear wheel trucks hauling campers on the highways today and the lack of news stories indicating that there are mass rollovers of said vehicles causing mayhem on the highways.
Yes, the fine details get into axle and tire ratings and staying under those is highly recommended. Trying to stay under the vehicles door tag weight will be an exercise in frustration and as others have noted, those door tags have more to do with taxes and registration fees than actual capacity.
However, it appears that hauling a horse trailer might also be in the works and that is a little different. Unless you are emotionally attached to the truck or financially limited, sell the truck and upgrade to a dually. What you want to do with towing horses and hauling a truck camper is doable on a single rear wheel but for me it always comes down to a tire blowout on the freeway. I feel much more comfortable driving down the freeway knowing I have the safety of a second tire should anything happen. Hauling animals in the trailer with significant emotional attachment is the tipping point in my calculations.
One suggestion though... If you are going to be hauling a horse trailer, consider having the tongue of the trailer modified so that it can be extended to meet the hitch of the truck with the camper on. I did this for my cargo trailer where I can extend the trailer tongue nearly four feet by just removing a pin and sliding the tongue out of the box frame. The advantage to this is that your don't need a super truss hitch which puts less bending moment stress on the frame of your truck, helps with ground clearance and sharp angle of attacks when hitting steep driveways and saves a lot of extra weight for not having a hitch extension.