Hi and welcome to the forum.
The Black Hills remains one of our favorite spots. For the type of site you've described, you might want to try Rafter J Bar Ranch - Home
It's close to Crazy Horse and an easy drive to everything else. If you'll be there for several days, consider watching the lighting ceremony at Mt Rushmore. If you have a toad that can handle rough dirt roads, maybe pack an evening picnic and head out to behind where they do the annual buffalo roundup in Custer State Park.
It's magical to watch the herds graze and as sunset approaches the coyotes begin to cry out and it sounds like there are hundreds of them echoing throughout the valley.
BTW, don't get too close. It is NOT a petting zoo.