We stayed at Buffalo Ridge Camp Resort in Custer, SD for almost 2 weeks and used that as a home base to travel to all the places you just mentioned. Buffalo Ridge is a Good Sam campground, so if you have a Good Sam membership, you get the discount price, and if I remember right, if you stay 6 nights, they give you the 7th night free. (That might have changed, but who knows).
While there, my daughter and her family also stayed there in a tent, but ended up sleeping in our camper more nights than not. The gradkids had a blast. We also rented one of the TeePees for one night (again the 10% discount) so the kids could sleep in the TeePee.
We were within a comfortable drive of the Bad Lands, Mount Rushmore, Custer State Park, Deadwood, and a bunch of other small towns. We took each day and did something different, and still didn't have time to take it all in. We were gone by 8:00 am and back at the camper about 10:00 pm ever night. Yes... we were exhausted, but OMG... so much fun.
We even went to Sturgess and visited the Full Throttle Saloon and the Camping World (brand new at the time) near Sturgess.
We brought home 2 bottles of wine... Calamity Jane, and Buffalo Bill wine from Red ASS winery and brought them home with us, which we still have, unopened.
We were very pleased to stay at Buffalo Ridge during the entire visit. Buffalo Ridge also has a train you can ride (free if you camp there), and great for the kids. They also have a dinner theater, which we never had time to go see because there was just too much to do everywhere else.
Buffalo Ridge ... click here.In addition, Buffalo ridge had a pretty good playground for the kids and a swimming pool and a game-recreation room adjoining the swimming pool, office. It has lots of acreage, and of course, full hook-up sites. Sites are a little close, but .... who actually camps there. The place is a ghost town during the day time as everyone is out and about, and in the evening, night everyone returns "homes". In spite of all the campers there, it was very quiet, and no hassles at all.
Shower rooms are individual shower rooms for family style units. They have cabins, the TeePees, and right on the edge of Custer, centrally located to everything.
Did I mention, we actually spent very little time at the campground, except one day we did stay put so the kids could swim all day in the pool, and we REALLY needed a "down day."
(and yes, our 1 ton dually made it through the tunnel on Needles highway.)