I have done iit 5 or 6 times, mostly in Suburbans or pick up trucks. The last time I went over the Beartooth was two years ago in a 36ft diesel pusher pulling a Jeep. It helped that I had been over the pass before so I knew what to expect. I did not need to use the brakes much, mostly just geared down and used the engine brake to slow down.
The only time I had any trouble getting over the pass was on a crossing I did one September about 20 years ago, ran into heavy snow near the top and had to drive in 4 wheel drive for about 10 miles.