If you were to insist on continuing with the Tundra, my recommendation would be to upgrade to a modern WD hitch with integrated sway control, such as a Reese Dual Cam or Equal-I-Zer brand.
Reason being is that these hitches will stiffen the pivot point between the truck and trailer and make them act more as one long straight vehicle when presented with side winds from passing vehicles. This will greatly reduce the push-pull feeling you are currently getting.
Even if you do get the 2500 truck, the WD hitch will not be a wasted purchase. It doesn't matter how heavy your tow vehicle is, your trailer is still on a ball behind it and having a modern integrated WD hitch and sway control to stiffen that connection while under way will make the tow much more enjoyable. Even if you don't really need weight distribution for the truck.