As a resident of WY, Yellowstone would be the last National Park that I would visit from Memorial Day weekend, till after Labor Day weekend. Most visited all of all parks, it's a train wreck. The magnitude of the douchebags you are competing with can be gauged by the amount of shit on their gubaru's luggage rack, they'll piss you off for the last 400 miles of your journey, no matter the route you take.
If hotpsots are a goal, make sure you don't have to holler and point something out to your wife and kids, you are about to be trampled by foreigners, and a 95lb Asian will most likely be sitting on your shoulders before the ordeal is over with. If you want reservations, make them now, both for lodging, and eating.
Instead of horseback trip, I'd find a helicopter tour, and try secure 3 full hours or what his fuel tank can withstand, as a guy above pointed, see more elsewhere, at cheaper rates. Horseback, you are paying for an experience, with some crap you may not want. Everything is at a premium from May 15th to Sept 15th.
I'm sounding like a real downer at the moment, but Deadwood nothing but a shitbox since gaming took over, old town Deadwood is long gone.
Mt Rushmore still cool, get a room in Keystone or Hill City, roll in around 4pm, grab a bite, go in the evening, see it in daylight, then stay till the lights come on. Badlands and Devils Tower always cool, as are other places on the route guys have mentioned.
I'd recommend Jellystone to anyone after Oct 1st till road closures, but if you can handle the people, go for it.