Having done the Beartooth in 2007 from Red Lodge, the Chief Joseph from Cody to the Northeast entrance in 2010 and in the opposite direction in 2014, I have to agree with others that towing such a long trailer over either route isn't a wise idea. Red Lodge is at 5,500 ft elevation and the Beartooth tops out at 11,000 feet, a long, long 5,500 ft climb. The Chief Joseph tops out at only 8,000 ft, but there are some pretty tight switchbacks going down from the pass toward Yellowstone.
Both routes are spectacular (we prefer the Chief Joseph for scenery), but are better experienced not towing.