The rear axle and wheel bearings on your truck is rated by the manufacturer at 10,900 lbs. and minus the weight of the rear of the truck at 3,000 lbs. that leaves a theoretical "payload" of 7,900 lbs. in terms of these components and based on every available seat in the cab having a 150 lb. passenger.
Ford like the other manufacturers calculates the payload for a truck based on the weakest link as that truck is configured at the factory. For every heavy duty truck I have looked at the load limitation has been load rating of the tires followed by the capacity of the leaf springs, both of which are easily upgraded. Easy to verify this for yourself with 10 minutes at a local Ford dealer looking at the different F-250 and F-350 trucks and looking at the payload rating shown in the glovebox.
If you think about what is different between towing 13,000 lbs. and putting 3300 lbs. in the bed in terms of the stress on the truck, it comes down to the frame, suspension mounts, leaf springs and the dampeners (shocks), the rims, and the tires.
I would bet the stock tires have a maximum load capacity for the rear pair of 6400 lbs. which after the truck weight is subtracted gives you the 3500 lb. payload figure.
Get a camper that has a published dry weight of 2200 lbs. or less and you will be fine with your truck as it is. It is likely the weight after extras like a battery are added will push the unloaded weight up another 300 lbs. and food, gear, and drinking water can easily add another 400 lbs. to put the total load at 2900 lbs. plus the weight of a hitch extension and the tongue weight of the trailer.
An option remains to upgrade the payload capacity of your truck by putting on higher load rated tires. I replaced my tires rated at 3195@80 PSI with one rated at 3750@80 PSI for a gain of 1110 lbs. of extra capacity at the rear axle. Going with 19.5 rims and tires would have increased the payload capacity even more. Adding Supersprings and possibly a rear anti-sway bar can help. All are commonly done mods on trucks to support the extra weight of campers.