The 262 should be within the tow specs for an F150 with a 5.4 so you should be able to pull it. Loaded it will probably weigh 6000# or a bit more, so it may be sluggish in the mountains. But I don’t know if that would be a deal breaker if your towing is predominately in the flat lands.
TT’s and FW’s both have their proponents and both work well for camping. The FW has about 20% of the total loaded weight on the pin so it soaks up the payload of a TV rapidly. So you need at least a ¾ ton TV to pull it and even a 1 ton if it’s a big FW. When I was a newbie I was thinking FW until I realized the pin weight. But the deal breaker was that it was too high to go down the forested roads I camp on. And I seem to remember one of your prior posts when you mentioned an aversion to steps, even in the shower and to a dinette slide. You realize it’s all steps to the bedroom in a FW?
The decision to go with a small TT worked great for me. I can get well into the back-country with it and having my F150 4x4 bed free expanded my options. I put a shell on it and had a mattress made for it. So I can load up some of my old camping gear and go even further back in and stay if I like.