I have a 3700 lb. camper when fully loaded and of that weight only 200 lbs. is carried by the front suspension, wheels, and tires. The diesel engine is carried by the front suspension, wheels, and tires, and adds no load at all to the rear axle and its wheel bearings and tires.
Focus on the rear of the truck. Tires are available for the stock rims with a load capacity of 3750 per tire or a total load capacity of 7500 lbs at the rear axle less the 3200 lb. weight of the truck which leaves a payload capacity of 4300 lbs. for the truck. That is the payload that the rear can handle with an adequate set of leaf springs.
GM has rear axles on its heavy duty trucks rated at 6000, 6500, 6700, 9300 lbs. and the difference is the leaf pack that is put on the truck at the factory. My 2500 SRW has a 6700 GAWR rating and standard 3500 SRW trucks have a rating of 6500 lbs., and DRW a rating of 9300 lbs. - and with GM diesel pickups the same axles, wheel bearings, and rear differential is used on all of them.
If the camper weight causes the truck to sag in the rear (with the new tires) then add SuperSprings. You can add one or two springs per side to provide a load capacity of 4,000 lbs. or 5,000 lbs. which is more than enough for most campers and gear.
Important to realize that the payload rating is calculated at the factory and based on the weakest link when the truck is assembled and shipped out. Otherwise identical trucks with different wheels or leaf packs can have vastly different payload ratings applied at the factory.
Modifications are done all the time which increase the payload rating as with adding Supersprings, new tires, airbags, and the like or to decrease the safe payload by lifting or lowering the truck or putting on flash rims and low profile tires.