Oh my Gawd! My dry weight on my trailer is 5,628 and I called StarCraft and they told me the hitch weight was 718. Are you saying that if I add 1,000 pounds inside the trailer that some of that 1,000 (12-15% of it) must also go on as hitch weight? Because if that is what you are saying then I would have to subtract 130 pounds more from the payload? Lets see...1580 (claimed payload) -150 (me)-120 (dogs)-130-718 (tongue weight). That leaves 462 and I don't even know what the receiver and ball weigh or what else I might put in the truck. I am kind of light on travel but still.
You'll be fine with the Tundra Double Cab. The payload rating should be around 1500lbs for that body style with options included.(check the door stickers for exact number) Since it's just you and some dogs in the truck you'll have plenty of payload left for the tongue weight, just load everything else in the trailer.
If the trailers dry weight is 5600lbs it'll be close to 6000lbs empty (as delivered with options). If you load 1000lbs of stuff into it (which is a lot for 1 person) you'll be up to 7000lbs. Well under what the Tundra can safely tow. At the average of 13% trailer weight on the tongue would give you around 900lbs. Call it another 100lbs for the weight of the hitch.
So 1580-150-120-900-100= 310lbs of payload left.
You can still add another 300+lbs before maxing out payload which is plenty to spare. You'll find similar results with most 1/2 ton trucks.