Mammoth will accommodate your rig just fine. It's first come, first served, so plan to arrive there before noon. Oh, and it is dry camping - water is available at spigots scattered throughout the campground, but there is no dump station.
We camped for five or six nights there in a 40' plus toad. The only issue was keeping the fridge lit - it didn't like the altitude much. We really enjoyed the campground - spacious sites, nice views, and Verizon cell/data work 5x5 there.
The other choice, route 26 over Togwotee Pass into Jackson Hole and Grand Teton National Park, is also a scenic drive with spectacular views of the Teton range on the west side of the pass. The climb and descent over the pass are both gradual. No white knuckle mountain driving. The park entrance fee for Grand Teton includes Yellowstone as well. You really shouldn't miss the Tetons when you are in the area. IMO the south entrance from Jackson Hole to Yellowstone is an easier climb than the east entrance from Cody.
IMO Grizzly RV Park in West Yellowstone is the best bet for a rig of your size.
Route 20 from West Yellowstone to Idaho Falls is a very easy drive.
We were there towards the end of last September, stayed at Grizzle Park in W.Yellowstone. I am glad I did, I had planned to stay in YS but Fishing Bridge closed by time we got there. After driving around with the toad, I am glad we stayed in W.Yellowstone and not have to drive the MH and Toad in the park roads.