The issue will be the payload capacity of the Tundra. The payload of 1640 lbs (7100 - 5460) will be with a 150 lb driver and without all options. By the time you add passengers, hitch and gear, that payload number will be closer to 1100 - 1200 lbs, maybe less. The pin weight of the camper using 20% of the dry weight will be 1580 lbs and of course loaded will be more.
I don't see how the Tundra's going to work out for you.
Edit: Another possible way is to calculate from the axle weight and the camper' GVWR. The manufacturer usually set the camper's GVWR by adding the dry pin weight to the axle weight ratings. As an example, 5200 lbs axles and a 1500 lb dry pin weight would result in a GVWR of 11900 lb. You should have the camper's GVWR and axle rating of the load rating label on the camper. The word of caution is that this would only estimate the dry pin weight...the loaded pin weight could easily be 300 - 400 lbs heavier.
Nights camped in 2013 - 55, 2014 - 105, 2015 - 63