I believe you may be confused regarding the Beartooth Drive. It is actually Montana Route 212 which runs between Red Lodge, through the Beartooth Mountains, through Cooke City and then into Yellowstone National Park at its NE corner. Beartooth Drive doesn't run into Cody, Wyoming. Route 14 in Wyoming runs from I-90, across the BigHorn Mountains, through Cody and then into Yellowstone at its middle, eastern entrance.
We've driven through the Bighorns with our 37', class A, pusher with a tow. Our route had several hairpins and some steep grades, but nothing to stress over. Slow and cautious driving is suggested, but it's a beautiful drive. Make sure you schedule your crossing when the roads have been cleared. Snow fall is possible even in late May.
I would not suggest taking an RV across the Beartooth Drive (Rt. 212) at any time of the year. That drive is way too demanding on our rigs, with lots more hairpins and steeper grades. There are very few turnouts and those behind you will not be very happy with you. We've driven it in our toad, but would never consider taking our coach across at any time of the year. This is the most beautiful drive in all of the Rockies!
Double check your map regarding which route you are planning to take.
Hope this helps.