We took the Beartooth Hwy route from Red Lodge to Yellowstone last September. We were hauling the truck camper. It's a steep and winding, but beautiful drive. We pulled off several times to enjoy the view.
I would recommend not taking the camper. The reason would be that you will want to be able to look around while driving and you will want to pull off every chance you get to see all of Gods beautiful creation. It will be harder to do this while pulling a camper.
Don't forget to take some jackets and hats. It's cold and windy on the top.
Below are some snapshots on the Beartooth Hwy from our trip last September.