Our favorite campground for the Tetons is Gros Ventre. We always drive right past the Ranger station and tour the loops to find the site(s) we like that don't have a dated occupied notice on the post. We then go back to the Ranger station and ask about availability of our #1 choice, which we already know is available. In the 2nd half of May the campground should be nearly empty, so you will have many good choices. There will be fewer choices on Memorial Day weekend.
At Colter Bay there is a sign-in station where you must stop to have a site assigned before entering the campground. Maybe just our bad luck, but there has only been one time when we didn't have to go back to the entrance station to ask for another site due to off level or too short for us to fit. FYI, the sites at Colter Bay are shallow pull offs along the sides of the loop road, similar to parallel parking. The sites at Gros Ventre are all back-in.
I wouldn't recommend towing beyond the first few miles of Grassy lake Road. We drove in as far as 10 miles in our 24ft Class C a few years back and should have turned around after maybe 3-4 miles. The further in we went, the worse the road got. That said, there were several nice looking campsites in the first few miles along the decent road.
2003 Winnebago Minnie 24F - Ford E-450🙂