The Tundra will work as is and I've loaded my basic shorty F150 heavier than you plan. You won't be able to carry as much as they advertise because you need to remember that the weights include you, your wife, the dog, groceries, and gas. Even so, this was about 950# in the bed and 1400# (about 300 tongue) in the trailer so I was technically overloaded with the biggest hits being body roll (not terrible) and gas mileage.
HOWEVER
While my truck already has the tow package that includes a heavier rear axle and tires, to tow a bigger heavier horse trailer I was going to put in air bags and decided instead to try the Loadmaster Adjustable Suspension kit for $400. (Now $450)
https://activesuspension.com
I can only say positive stuff about it for the money and maybe 45 minutes total that it took to install both sides. It's much better than helper springs and the airbags I've had before on other trucks.
The unloaded truck takes bumps MUCH more smoothly and there's less sway in corners. It's definitely better than bags I've used in the past, plus this will never develop an air leak. Unlike regular helper springs the coils are progressive rate so it doesn't just stiffen the rear. The more sway in a corner, the stiffer the springs get on the low side and the truck just goes around flatter and in better control. Plus, the more you load the rear with cargo the stiffer the springing gets. And if I really want to haul something heavy like that horse trailer, I just need to turn the all-thread core rod with a ratchet to stiffen the rear.