Did a similar trip from Indiana to Custer, SD. We stayed at
Buffalo Ridge, which is a Good Sam campground.
We took 3 days traveling to get there, 2 days to get home. En route, we did overnights at KOA's. But we did have to book all reservations months ahead of time.
We stopped in and saw the Corn Palace in Mitchell, SD on our way there.
We took 2 weeks, so basically had 9 days in the area. My daughter and husband and 4 kids also came from Kansas City, Mo and tent camped. We foot the bill and rented one of the Tee-Pees and let everyone sleep in it, which was an experience to say the least.
From Custer, I felt we were pretty centrally located to scurry off and make day trips to all the locations you've mentioned above. We did not go to Wall, no desire to see it, but we did hit the Full Throttle Saloon in Sturges. Saw Deadwood, Mount Rushmore, Crazy Horse, all that stuff.
Because we had the grandkids with us, we also went to that animal zoo thing, and that candy store not far from Sturges. (don't remember the names now). We also built in a couple days where we just stayed at the campground.
Buffalo Ridge campground has a train you can ride (kind of fun), and a dinner theater (which we simply never had time to do), in addition to swimming pool and play ground for the kids. Being our age, we needed some in-activity every other day. In July it was hot and steamy, never rained, and did I say hot. From what the locals had to say, that was an unusually hot time of year.
We also traveled the Badlands, and got into the Black Hills. We didn't do any ATV rentals, but they are everywhere. I wish we would have had another week there, there is just so much to do.
Personally, I'd take the camper, select one campground near the Custer area, or Custer State park if you can get a site and use it as a home base the time you are there.
FYI, if you want a reservation anywhere in that area, start calling now! Find out when reservations can be made for the anticpated route of travel, and at the end destination point. If it's too early to make reservations now for next year, then find out when the first day is open for reservations at each location you want to overnight at or the final destination. In July, absolutely everything will be 100% booked solid.