All the manufacturers calculate a payload capacity based on the expected passenger load (why as mentioned before that a regular cab truck will have a higher rating than an extended or crew cab truck that is otherwise identical) and the weakest link (suspension, frame, axle, rims, tires). With the heavy duty trucks the weakest link is the tires which is why there are DRW versions so there are twice as many tires to support the load in the bed. The rear axles used by Ford and GM on their heavy duty trucks are rated at 10,900 lbs. and subtracting the rear axle load from the truck when empty that leaves 8,000 lbs. of load capacity if only the axle and its wheel bearings the limiting factors.
Most likely your two rear tires have a total load capacity of less than 6500 lbs. and that is what is limiting your truck's payload capacity. There are GM heavy duty trucks with trick 20" "offroad" rims and tires that have stickers in the glove box stating that the trucks are not to be used with a camper. Same truck with different rims and tires have payload capacities up to 2800 lbs. as with my 2011 Chevrolet 2500HD extended cab short bed truck with the Duramax diesel engine.
I added a set of SuperSprings for $445 and replaced the stock tires with Nitto Terra Grappler 285/70R17 tires rated for 3750@80 PSI. The truck easily handles a 4,000 lb. fully loaded camper in the bed and the tires have a 500 lb. margin in terms of their load capacity. Any heavier and I would be looking at 19.5" rims and tires or going to a F-550 monster truck or Class B+ motorhome.