A friend of mine lives near Victor. He says under no circumstances try to take a trailer of any kind over Teton Pass--it is widely known for being a transmission and brake killer. There are 10% grades on both sides.
There are numerous stories of disaster on this forum from those who ignored the warnings!
A bit more of a drive--but much safer and cheaper in the long run--is to take US 89 down through Hoback and then US 26 up to SR 31. I've been that route and it is very scenic.
You can also go over to West Yellowstone and down US 20 and SR 32.
INFO FROM WYOMING DOT ON TETON PASS