What can I tow with a Tundra DC?

Stick to around 6k dry and you should be pleased.

A look at your cargo handling label on the front driver's side door pillar will be a good first step. With smaller vehicles, the weight put on the vehicle, the trailer's tongue weight, is the most important. That weight, along with any passenger and cargo weight, will describe the total capacity of the vehicle while towing.
There may also be weight limits on the vehicle's towing capacity (weight on the rear end), intrinsic limits due to comfort and towability, or other reasons not to exceed a certain amount of weight. Your floorplan choices are probably going to push the limit of the Tundra's towing configuration but there are many happy Tundra users that haul travel trailers. I would expect that trailers with a weight of around 5000 lbs, unloaded, would be towable with a Tundra.

Here is a specifications table that lists the weight handling of the Tundra's different models: Toyota Tundra link

You may also wish to check Toyota Tundra Forums where really applicable towing methods are discussed. Towing a large sail down the road is sometimes beyond just the weight numbers.