Just brought a new Chev Silvarado 1500 with 20" Goodyear Eagle tires. When doing a little research on the tires, which I wish I had done before buying the truck, found out they're passenger car tires....
Your truck tires have plenty of load rating for your truck, even towing a TT. Just keep using them. Going to LT for a 20" wheel will require a slightly larger tire size. That plus increased weight of an LT tire over P will decrease performance, acceleration, braking, and fuel economy.