I'll second the "Go for it"!!!
Beartooth highway is a "Must See" item on your tour. Along with Warpath from the site, we were just out there in June this year and drove the highway from Red Lodge MT to the North East entrance of Yellowstone.
We made the drive without any problems in our 36' pusher towing a car. The pass is long and sections of it are moderately steep, but certainly driveable. The most difficult sections of the pass are due to the switchbacks rather than overall steepness. The turns are tight and with our rig's overall length approach 60 feet with the towbar and car, we did have to use all the road for swinging wide on the switchbacks.
We saw a few travel trailers, and other truck campers while driving the pass. Providing you've got a well set up camper, you should run the pass. You'll really enjoy it. If you do have any concerns while running it, there are plenty of pull-offs available to be used by slower vehicles and you can take a break along the way.
You can select my photo album to see more pictures, but I thought I'd post a couple here so you can see the actual beartooth highway.
This picture is of our rigs at one of the pull-offs while climbing the beartooths.
Decending the Beartooth highway.
It's hard to see, but at the bottom of the photo towards the right is the highway then again on the left side is the same highway carved into the side of the mountain.