That size and weight is perfect for the Tundra.
There is no Tundra with an 11,200lbs tow rating anymore under the new SAE standards. Not sure where you got that number from. The highest available is a regular cab 2x4 at 10,400. With 1500lbs payload as you say, I would suspect yours is a double cab model? It's tow rating would be 10,100lbs.
You were given incorrect info on the hitch as well. The receiver is rated for 500lbs without a wd hitch. It is good for up to 10 percent of its tow rating with one.(1,010lbs).
If you plan a lot of x-country trips, upgrade the tires to E rated ones if not already equipped. The stock P rated ones are pretty soft. Any good wd hitch with built in sway control (ie Equil-i-zer, reese dual cam etc) will work just fine.