The only campground I am familiar with is in the Badlands.
Welcome to the Badlands Budgethost Motel and Campground
It would be a good place to stay while on your way to the Mount Rushmore.
The Badlands loop road in a nice very scenic road - well worth your time. Easily traveled with trailer but may be nicer to tour with just the tow vehicle as there are a lot of scenic turnouts.
If going towards Rapid City from the town of Interior use Hwy 44. It is another lightly traveled and scenic nice road. Definitely see Custer State Park - nice drive but do not take your trailer as there are a few tunnels.