If the truck truly weighs 4088 empty, and has a GVWR of 6030, then he has 1942 lbs of payload/passenger/fuel capacity in the truck.
That's pretty impressive for a compact pickup.
If the tongue weight of the trailer, plus passengers, plus fuel, is less than 1342 pounds, then he indeed does have 600lbs of "stuff" room in the truck. HOWEVER, I would highly recommend against taking advantage of that capacity because it puts you right at the limit.
The trailer, loaded to the max, will probably have a tongue weight close to 1065 pounds. Right on the back bumper of that little truck, you'll be the only "wheelie" camper in the RV park, even with weight distribution. More than likely you will be exceeding the rear axle weight rating GRAWR.
KE5DFR
Vintage 1979 Silver Streak Supreme Rocket toted by a 2002 F350, crewcab dually, 7.3L,4.10 axle,SCMT. Travel with two miniature Schnauzers and one African Gray parrot. Practicing for retirement!
First off...the curb weight of the Tundra seems low, considering mine weighs about 5,500 lbs and is a 4X2. newer model, but I doubt there is that much difference between the two.
That being said, the Starcraft you are looking at shows dry weights on their website. That trailer with options will weigh about 6K lbs. Add propane, a pair of batteries, water, etc. and the unit will weigh over 6,500 lbs. By the time you pack yourselves in the truck, add food and all the other "things" you bring with....bingo you are at maximum towing weight.
Not saying I have not seen this done before, but I doubt it would be enjoyable. Good luck in your trailer searches if you decide against this one.