So Confused

APT wrote:
While payload is often the limit for half tons, I think it will be acceleration and power for your truck.

Concur.

Not sure what the HP output of the OP's engine is, but 5700 lb tow capacity is what it is. Our own BTDT experience:

We used to tow a 20' TT with a loaded weight of 5000 lbs with an F150 with a V6 (non-ecoboost). It was rated to tow 5500 lbs, not far off the OP's capacity. The TT had a UVW of 3800 lbs and ended up weighing 5000 lbs after all mods and loaded up for camping.

The combination towed like a pig. It was slooow to accelerate. Driving into strong headwinds was awful - it could not maintain a steady speed. Worst of all, on steep hills, it could only make 30 - 35 mph flat out, pedal to the metal. On the interstate on steep sections, we were on the right shoulder with the semis crawling up as low as 30 mph. The engine was always running at high rpms to make enough power. Gas mileage was terrible and as low as 5.6 mpg. On an 1800 mile trip, we had to stop at 3 Ford dealerships to get engine work done because it was so over-worked. The down-time put a big damper on the trip too.

Don't want to rain on someone's parade, but I see several options IMO - only go on short local trips and avoid hills; get a very small & light TT or pop-up; upgrade the truck. If the OP intends to get into frequent camping, I would upgrade the truck and even go to a 3/4 ton like we did. Our 20' TT (brand new) only lasted 1 season and we upgraded to a 29 footer along with a 3/4 ton truck. Now on our 2nd season with a 3/4 ton (and 3rd TT) and love it. We even get better gas mileage with a V10 and heavier truck and heavier TT.

Regardless of a truck and engine's spec's and payload capacity, towing near the max. tow capacity is just not a great idea.