If you take the Going-To-The-Sun Road, I recommend doubling back on Highway 2 to Essex, Montana, one of the most charming places in the world (IMHO), check out the old railroad hotel, the railroad operations, the Goat Lick, the Middle Fork of the Flathead River, you name it. Things don't happen too fast in this part of the world. It's been thirty years since I've spent a lot of time there, but I swear the same potholes that were in the road then are still there.
You can't travel Going to the Sun Road with a trailer! Nothing over 21ft................. So no matter what you will have to go hwy 2 , then if you want to see The Sun road, drop the trailer and see it. Enjoy!
I would also suggest when coming thru Montana, plan your stay on the West Side versus the East Side. More Senic, Cooler, and in my opinion nicer campgrounds.