The gross vehicle weight is a factory calculation and largely irrelevant. 95% or more of the camper load is on the rear axles and its wheels and tires. If you have two tires at the rear rated at 3200@80 PSI the maximum load is going to be 6400 lbs. minus the 3,000+ lb. weight of the truck itself.
The payload rating sticker in the glove box provides information as to what the factory provided leaf springs can support. If it is enough then fine, if not then add springs, i.e. Supersprings.
There are tires from BFG, Toyo, Nitto and possibly other companies that are rated at 3750 or higher and will provide an extra 1100 lbs. of load capacity at the rear axle.
Use a local CAT scale and weight the truck with and without the camper. With the actual weights at the rear axles loaded and unloaded you will have the information you need to make choices as to modifications needed.
Most likely with your F-350 the current weakest link is the tires which can be upgraded. The higher load rated tires will also reduce sway with their stiffer sidewalls which is a bonus.
The camper you mentioned is at the lower end of the weight scale for hard side campers and your truck should be able to easily handle the load and its high center of gravity.
Many trucks also benefit from the addition of an after market anti-sway bar at the rear.