RE: Renting a Tow Car.
One might check into renting a car in one of the close by towns, such as West Yellowstone or Jackson. If you have someone able to drive the rental car & one the Class C, it would work fairly easily. In doing a cursory check there were several rental car sites in W Yellowstone & Jackson both.
One could rent on in each town to use in touring the park near them. Or rent one in one town & drop it off in the other. Maybe.
We had a small Class C for about 10 yrs. From '97 to 2005. They are somewhat easy to set up & take down. Not too hard to drive around to see things (never took it to Yellowstone), usually had a tow with it, but did do solo once or twice.
Traffic in Yellowstone can get pretty tight, especially in the pullouts (parking lots) for exhibits & sights. Saw someone take out a side mirror at one. At times they have very large tour buses pull in as well. But with a little patience & a lot of caution, one could see most of the 'drive by sights' without much trouble.
Always see many rental RVs in the campgrounds & on the roads up there.
If having to drive the MH in each day, IMHO, I'd just as soon camp inside the park. One could chance campgrounds once or twice as well inside the park. Then you wouldn't have to recross the same route each day.
If having to 'de-camp' each morning anyway, it wouldn't take much time to dump the gray tank & fill with fresh water. Driving would keep the batteries topped off from using them to dry camp.
If needing TV or wifi, this wouldn't work. Not much if any in the Park itself. Might have TV in Tetons & find a hot spot for wifi there.
If touring with the MH, you will always have your own bathroom & eating area with you & your own fridge. Ha! One could even take an afternoon nap, before going to the next site to see what's happening.
Any of the campgr'ds in the park, such as Grant, Canyon, Bridge Bay, Mammoth, Madison etc., would be ideal. It is a big place takes some serious driving to see the place. Much easier to do so in a more central location. Drop down to the Tetons before you leave, it is neat and somewhat different.
This
Site Clicky tells about each campground (down by "Click on a Campground Name Below to Take an Interactive Tour"), if the chart has DS for Dump Station in the Features column it has a dump station. Ha! Grant Village, Canyon, Bridge Bay, Madison & Fishing Bridge have them. One could use Mammoth or Norris (ones a generator can be used) & still dump in one of the other places.
In the Tetons: Gros Ventre Campg'd or Colter Bay Campg'd. We stayed at Colter Bay Campground, the one without hook-ups in mid-June 2010 and once before some years ago. The Colter Bay Visitor Center and Indian ArtsMuseum by the Colter Bay Campground is really neat.
I have heard good things about Gros Ventre Campground...get more animals walking by or close by, etc. Nice views. It seems to be named after me as well, (from French: big belly). Actually, the French used the term Gros Ventre, for a local native population/people which was mistakenly interpreted from their sign language. Instead, the Gros Ventre people call themselves by the autonym A'ani or A'aninin, which means "white clay people," or more loosely "ourselves, our people, us, we," etc. The French is pronounced "Grow Vaunt."
Both Gros Ventre & Colter Bay have dumpsites.
We stayed at Mammoth Campground for 2009 visit to Yellowstone. It has some very large spaces in it, almost any size RV could fit. But can't make reservations, but we have never done so in the four visits we have gone there over many years.
We stayed at Grant in 2010, pretty large campground, all sizes of RVs. Bridge Bay looked like it would be a nice one to stay. We did stay there in '86.