Your truck can handle the camper load with no problems. It would be a lot better with 3.73 or 4.10 gears and I would not buy a truck with 3.55 gears which are fine for a SUV that will only handle a passenger load and not be used for towing.
I would guess that the truck itself puts a load on the rear tires of 3200 lbs. and add in 3700 lbs. for the camper fully loaded gives a total load on each tire of 3450 lbs. which is probably over the max load rating for the tires on the truck. This is easy to upgrade with Nitto or Toyo tires that will fit your truck's wheels and provide 3750 lbs. per tire or 7500 lbs. for the pair at the rear axle.
Once you have the camper on the truck go to a CAT scale and check the load on the rear tires. Well worth the $10 cost. Weigh it again when the camper is fully loaded for your first trip.
Something that will help your truck is the addition of a rear anti-sway bar if you did not get the factory option with your truck. Hopefully you got the tow package option as that adds extra cooling capacity for the transmission which is important with a camper load.
If the truck sags in the rear, which is unlikely, then you can easily add a set of SuperSprings (as them what they would recommend in your situation) to gain more payload capacity.
If the springs are up to the task then the other possible weak point is the shocks. The factory shocks are not likely to last very long with the camper load and when they do need to be replaced go with Bilsteins or Rancho 9000XL shocks.
Your truck's rear axle and wheel bearings are rated for more than 11,000 lbs. and the truck's drivetrain and frame is rated for towing half again that amount so don't worry about hauling a load of less than 4,000 lbs. in the bed of the truck.