You are NOT
supposed to take this route! I found it years ago and fell in love with it. Now everyone wants to drive it. Please don't go there because I'm trying to save it all for myself.
Seriously, this route is awesome. It's the ONLY way to enter Yellowstone, but it is different and you need to know how to deal with it. It has some switchbacks and very steep climbs but it's manageable and the views are awesome and definitely worth it. We took this route in June of 2003 (see the picture below at West Summit) with our Suncruiser 33V with W20 chassis and had no problems handling it. We tried it two other times in September but got shut down by blizzards and couldn't go through. This last September we made it through half by sneaking in the back way and bypassed the snow. If you are traveling in June, July, or August you'll be fine.
First, you will leave Red Lodge and after 14 miles of a gradual but steady climb through the fores you will begin a series of hairpin switchbacks. This is the toughest part of the trip. My Suncruiser rarely got over 35 MPH on the tough part because the straightaways quickly gave way to another hairpin so we had to back down to 15-25 mph for the curve and then nail it again. We unhooked the Cherokee for these switchbacks, which helped a bit on the power, but mainly it gave us a chance to fit into the pullovers when traffic got behind us. It'll be easier to park your 35U in a slot than it will the whole train. Once you are up at the first scenic overlook and rest area parking lot (6 miles later) you can rehook. You will want to stop a number of times on the way up because there are some awesome overlooks that you won't want to miss. I hope you have a digital camera because you'll go broke if you are shooting film on this trip. There's a reason the Charles Kuralt called it the most scenic road in North America.
You will see frozen lakes, a ski area (yes, they'll still be skiing in June), frozen tundra and permafrost, streams, wildlife, etc. After your about 30 miles out of Red Lodge you will come to the West Summit, which is around 11,000'. Look around and you'll see the Beartooth and you'll be looking DOWN on snow capped mountain peaks. After a number of great stops at scenic pullovers you'll eventually get to Top of the World Store, which is 39.3 miles from the convenience store at the south end of Red Lodge. Be sure to stop here and check it out. You can actually camp here in summer. They have no electricity but there is a sewer and water hookup. We also had a small grizzly underneath our coach at night but that's another story. Unhook the toad and go 1.8 miles down the road to the Beartooth Lake area. Check out the lake and the view of Beartooth Butte right across the lake (by now you're on your 3rd roll of film
) then go down the road a piece and you'll see a waterfall on your left as it empties into a canyon. We scrambled down to the base of the waterfall but at 10,000' altitude it'll be work climbing back out of the canyon if you do. Then rehook the toad back at the store and continue on down the mountains. There's plenty of more things to see on the way down to Cooke City, such as Pilot Peak, Sunlight Basin, and Crazy Creek Falls so now you can burn another roll of film up.
Going down into Yellowstone is a more gradual and longer grade so brakes are not an issue. If you leave Yellowstone towards Red Lodge you will have to deal with the brakes on those switchbacks so I don't recommend it. It's much easier to do the hard work when climbing, not descending. Besides, entering Yellowstone this way allows you to build your enjoyment as it just keeps getting better and better.
If the weather turns wrong, you have three options. If it's big time bad you can go back out to I-90 and head over to Bozeman and then down through the Gallatin Canyon to West Yellowstone. If it's not so bad, you can get to Cody from Red Lodge then enter the east entrance. Last September, they had snow storms on the Red Lodge side, which is common. There are two summits and Top of the World is in a high saddle between them and the Cooke city side generally doesn't get snow as often as the Red Lodge side. We then drove out from Red Lodge to the east and then south through Dead Indian Pass and Sunlight Basin via the Chief Joseph Highway. This allowed us to enter the Beartooth Highway at it's midpoint, just down range of Top of the World. So we still got into Yellowstone via the Northeast Entrance.
Also, at this time Dunraven Pass is closed for road work. To get to Fishing Bridge you'll have to go through Tower, then Mammoth, Norris, Madison, and then Canyon to get there. Be sure to allow the whole day to do this. It's only 68/7 miles from the C Store in Red Lodge to the East Entrance station at Yellowstone but there's so much to see and do, it will take some time.
In summary, your coach can handle it, you need to allocate plenty of time, and the weather will be a major player, especially if it's not one of the 3 summer months. But, it's all worth it.
Another consideration - Yellowstone is a big park. If you try to see it all from one location you'll spend much of your day driving back and forth. We generally split our stay. I'd consider staying at Grizzly RV Park in West Yellowstone first, then moving to Fishing ridge to see that half of the park after a few days. It would be a better utilization of your time.
Enjoy the trip.
JUST JUST DON'T TELL ANYBODY ABOUT THIS ROUTE!!!