I tow a very similar trailer to the one you do with the Tundra. It is a very capable tow vehicle and I would not hesitate to tow the trailer that you are currently thinking of towing with a Tundra. A couple of points to make.
Real life gas mileage have a comparison to the F150 and the Tundra on Fuelly.com and there is very little difference in real life from people entering their mileage between the F150 and Tundra or any of the other 1/2 ton trucks.
Payload rating Toyota is very conservative. If I compare my Tundra to a 2006 Sierra (not an F150 I know) but the big 3 rate very similarily on this, king of the hill mentality. Sierra 1700 lbs rating, with 1700 lbs in the box 1 inch clearance on the end stops. Tundra 1290 lbs rating, 2000 lbs in the box 1 inch clearance on the end stops. Would I put 2000 lbs in the Tundra all the time, no, but using the same criteria to load and haul junk to the dump the Tundra handled it better.
The Tundra with the 5.7 engine will also tow your trailer much better than the F150 with the 5.0. As others have stated the Tundra 5.7 is more like the F150 6.2. I would not tow my trailer with the 5.0 F150, just not enough power there.
Just my 0.02. Drive both and pick the one you like best. I did and am very happy with my choice.