If you had selected a F-350 SRW, it would have about the same tires and axle ratings. However the GVWR is 1,500 pounds higher on the F-350 SRW than the F-250 SRW. Simply put, you have the wrong truck for a 1 ton load, such as a fifth wheel, and be able to carry passengers as well.
You can tow most any travel trailer, with it's 10% - 15% hitch weight rating. Even a 12,000 pound travel trailer with a 1,500 pound hitch will still leave you 500 - 600 pounds of weight capacity for the weight of the hitch and passengers.
With a fifth wheel, the hitch alone comes in a 150 pound crate. The hitch pin weight when full of water might come in around 2,000 pounds, and then you expect to carry more than the driver, so add that weight.
Check the curb weight of your truck. It might be lighter than you expected, and subtract that from your 10,000 GVWR. This will tell you how many pounds of passengers and hitch weight you can actually carry.
Just don't decide to also install a extra large fuel tank and carry 500 pounds of firewood too! If you are over the GVWR by say 200 pounds, it is not the end of the world.
If you are confusing the 3,750 pound tire weight rating with the ability of the truck to tow 14,000 pounds, remember that most of the trailer weight will be on it's two skinny axles, and china bomb tires. So you only need to worry about the weight on the truck's tires, and axles, and the brakes can handle.
Good luck!
Have fun camping,
Fred.