Tow later - the dance starts - need new TV for the TT, then can get a nicer TT, then need new TV again, and on and on it goes :)
Don't ask how I know...
BTW - Towing: makes sure the nose is dead level to slightly down (slightly down tows better for us), and measure front of car wheel well height without TT on and use weight distribution bars to redistribute weight such that you're near that same height with TT attached.
BTW2 - Tranny temp: I can't guarantee, but depending on how hot is was outside your temp doesn't sound too bad to me towing flat, BUT I expect you will probably have issues with mountains.
Good luck, and happy TV hunting (I just bought a Silverado 3500 diesel, our 3rd TV...)