I grew up in Wyoming and now live in Colorado. I try to visit Yellowstone every other year...although I've gone there twice in the same year a couple times. If I've driven WY22 it is so long ago that I don't remember but most folks say avoid it in an RV.
The other routes I have driven. I would not take either of the routes mentioned. See this MAP
. I would take US 287 from Rawlins all the way to West Yellowstone. Note: US 287, US 26, US 191, etc are the same highway at various points on this route. It is the shortest and I suspect takes less time in an RV than either of the other routes.
Between Dubois and Moran the highway traverses Togwotee (pronounced Toe'-gu-tee) Pass. It is a good climb but easily traversed with any properly loaded MH built since 1980. I make this drive in a gas 34' MH with a Jeep in tow. Previously I did it in a gas 1988 32' MH and before that with a 1972 Ford F250 hauling a 12' camper.
Once inside Yellowstone the route has some relatively short 6% grades up and down. You will probably have more problems with wildlife holdups than with curves or grades. The highways are GOOD.