I would also have you consider some other campgrounds in and out of the state park. Bluebell is much nicer than Legion in the park. Another good park outside of the park in Custer is Big Pines. If you like off beaten path, try the dirt roads on the wildlife loop. There are quite a few and the one that ends up back at Bluebell and passes the horse park is fun. We were just there staying in the Super 8 as we have property there and saw a campground called Fort Wellikit. It looked pretty nice. And kid friendly.
Be sure to get one of the "Black Hills Maps." They are at every checkout in most places you might shop. You can find one here. https://www.blackhillsvacations.com/plan-your-visit/black-hills-maps
As for a backway to the Badlands, you can do it. In fact, one of the side roads out of the Wildlife loop in the park heads toward 79 and then take 40 and then there is one road that will head over toward the entrance. Heading out early in the morning and doing the wildlife loop then is good and you could just head over for a nice adventure. You can't get too lost. IMHO Custer (the town) is the best base camp area. It will cost 20 bucks for a week pass in the park (per vehicle). The Black Hills is wonderful. Be sure and do Iron Mountain Road (pigtail turns0 and the Needles Highway. They are musts. Rushmore well you do have to check it off your list but once youve seen it you are done. Iron Mountain and Needles have these great tunnels that as yo go through them, you can see Rushmore from a distance. Neat.
Places to eat. Bugling Bull Custer, The Alpine Inn (real German food for lunch) in Hill City. Sylvan Lodge for a drink by the fire...well maybe not in June.