OP, thank for posting number of people. From your posts it does not look like you will be pulling a car. In the peak season at Ystone you are going to have trouble finding parking at many of the pullouts/sites with a 31' MH. Most of the side roads will not allow you on them except in a car or PU.
You have posted that you are a FHU person BUT I would suggest staying at Baker's Hole USFS CG just north of West Yellowstone and renting a car to see the north side of the park. $21/day with electric will help pay for some of the car rental. If you are into long hikes each day then having a car is not as important.
You can park your rig at Old Faithful without problems and then I would proceed on to Grant CG for one night. Not as many pullouts or side roads on the south side so doing it in the MH will be easier. I would give the north side three full days and then south side one day. That will put you in the Tetons before noon giving you 2 1/2 days there if I read your timeline correctly.
Seeing the Tetons and the town of Jackson will be OK in just the MH. 26/287 is a good road out of the Tetons and it's 500 miles to Rushmore.
I'll assume there is no way to make this a longer trip so if you can do 500 miles in one day then I'd do the 1200 miles from Rushmore home in two days giving you another day in Yellowstone/Tetons.
I would probably go to the Tetons first as getting there will be a better route. Granted it will be a little further from West Yellowstone to Rushmore (550 miles) but on better roads.
Are you sure you cannot get more time for this trip? Hope this helps.
