The tires support 7500 lbs. and subtracting the 3340 lb. weight of the truck that leaves a load capacity of 4160 lbs. for the camper. The rear axles can support at least 8900 lbs. on a Ford truck and newer model years can support even more weight.
The difference between the 4160 and the "payload" figure from Ford is due to the limitations of the leaf springs provided. Best way to augment the factory springs is with a set of Supersprings for $450.
Sumo Springs are not a good solution as they do not work like the factory springs and they are only warrantied to last for 2 years. Aftermarket suspension mods that limit the range of travel can work but the do not truly increase the load capacity of the truck. Only more leaf springs can do that without affecting the ride and handling of the truck.
Supersprings makes sets to provide 4,000 lbs. or 5,000 lbs. of total load capacity and I would go with the higher capacity. When turning on slopes more than 50% of the weight is on the "downhill" side of the truck and the leaf springs on that side.
My camper weighs up to 3700 lbs. when loaded for a trip and less than 200 lbs. is carried by the front wheels with more than 3500 lbs. being carried by the rear axle and wheels.