My 2005 F250 with camper package carried a 4000 lb TC while towing an 8000 lb enclosed trailer, but it required 19.5 wheels, shocks, upper and lower StableLoads, airbags and SuperHitch with 28" SuperTruss. It was a very steady setup and traveled most of the Pacific coast and as far east as the Mississippi River.
If you plan on keeping the truck for an extended period of time, the upgrades are worthwhile. If you think you will be getting rid of the truck in the next three years, the cost is hard to justify and it would be better to upgrade the truck and then save for your next TC.
BTW: Your numbers seem low for FAWR and GVWR. I know my Super Cab short bed had higher ratings.