You should be able to find a 13-15 F150 that has the Max Tow pkg. The Max Tow optioned 150 was much more plentiful than the HD payload pkg. If you get the crew cab, it should have 7,600 GVWR. The max tow should come with 3.73 in those years, as well as the large tow mirrors, and IBC.
Your TT would be the limit for most F150s, but you being an experienced tower, it would be doable IMO. Yes, for towing a 250 would be better, if you plan to tow a lot. If you and wife plan to use it mostly as a car, she will enjoy the 150, which will not only ride much better, but better fuel mileage in every day use.
Jerry