We towed a 26TBUD with our Tundra. It had the power but it felt like we were towing a barn back there. No idea why, it just did. Ended up taking it back to the dealer after numerous problems with the axles and other parts and bought our Surveyor. Its longer but lighter and tows so much easier.
My yellow sticker shipped weight was 5900# on our 2012 TBUD. If you are adding in another 1400# that is not exactly travelling light! You are also looking at a mid 900# tongue weight. Add in 700# for you and the dog and you are at 1600# in payload weight plus the stuff in the bed, hitch weight, etc. In other words, you will definitely be pushing the trucks payload. Engine will have plenty of power to pull it around.
I'd pay the small upgrade to get the HD Payload package and another few hundred pounds of payload if you havent yet bought the truck (and if you are set on the F150).
BTW, the layout of that TBUD is really nice. It actually felt roomier than our longer TT with that big slide and open layout.
