The mid-size pickups can handle it, with the caveat that you plan on driving about 60 mph (not much faster) most of the time, and in the mountains drive 35-40 mph up the long uphill grades. If you want to run around faster than that (and use more fuel) then the 2.7L turbo F150 or 3L Ram turbodiesel will be more satisfying. Not just stronger engines in the larger pickups, but more cooling capacity and heavier duty transmissions, too. But yes, the mid size pickups can do it if you don't push them hard all the time. Realize that you're dragging a huge sail behind you, and a small increase in highway speed causes a large increase in power consumption. (BTW, non-turbos lose something like 2% of their HP for every 1000' of elevation gain, so turbos have an advantage in the mountains where you need power the most.)
Personally, I would want at least 1000 lbs of listed cargo capacity (CCC). Water is 8.3 lbs/gallon, and you'll have 6 gallons in the water heater. If you fill the fresh tank, that's another 300 lbs. Add in LP, battery, and gear. Be sure to locate and look at the actual-weight sticker (usually inside a cupboard door) of the unit you're thinking of taking possession of, to see what CCC you really have after all options, then add the weight of the other items mentioned above to get a better picture. And all that will be riding on a single pair of tires; check the load capacity of those on the TT.
Not trying to discourage you, but go in with both eyes open and well armed with info.