That 785 hitch weight is dry. I'll guess it'll be closer to 1000 lbs loaded. Figure another 50+ lbs for a WDH you are now at 1050 lbs. I'm guessing your available payload is around 1400 lbs in a crewmax (assume that's what you meant by maxcab). You are going to be pushing, if not over, your payload/GVWR.
If you don't mind the possibility that you might be over you GVWR, you should be fine. The engine has gobs of power. With a quality WDH and sway control you should be able to handle a 31 foot RV... but that is pretty long for a half ton truck.
You probably have some spongy P rated tires on your Tundra. You might look into upgrading to LT tires for less lateral movement while towing. My Tundra has E rated tires from the factory. They are very solid and don't give me any feelings of sway while towing.