You can certainly do your planned trip. However, you mentioned trying to visit the geyser areas while you have the 5th wheel attached. The problem with that is lack of parking spaces. You should plan to tour after you drop the RV.
Take the Red Bus tour of Going to the Sun Rd. You'll all enjoy it a lot especially if they can put the top down! It's very informative. Skip eating out a few times to save up for it! It's worth it.
Some schools in the West are over in May but I think your June trip still will be without a mass of tourists. Mid-June to Mid-August are the big times. Weekdays don't matter. When folks have a 2-week vacation many will be there during the week.
If you need to be flexible with Cody then you might consider Jackson, WY's rodeo on Wed & Sat. while you visit Grand Teton Nat'l Park. I doubt if your daughter would like Cody's museum unless she's into western history. BTW.... Jackson has some large grocery stores if you need to stock up and good fuel prices south of town. Colter Bay RV park if you need hookups or Gros Ventre campground is an excellent location if you can do without hookups. It has a dump & water fill. Can't reserve but we've always got a site in various seasons.... 300 sites. Both are in the national park.
You said you're going home via Michigan. Try to stay around Sleeping Bear Dunes National Lakeshore. That whole area is gorgeous. Take the gentle canoe/kayak trip down the Platte River which will end at Lake Michigan - very nice. It's a gorgeous area.
This is a great trip for your daughter. Take it with positive thoughts and you'll enjoy it. She'll have good memories!