first thing to know: The Sturgis Bike Rally will be held from Aug 4 through 13, which means a whole heck of a lot of motorcycles will be filling up the Black Hills campgrounds and motels, etc.
I don't know of any boondock places. Since you're coming from the west on I 90,
I suggest you take the Moorcroft exit (highway 16) and go to Custer, SD. Then continue west on 16 to Custer State Park and drive through the park to other side where there is a nice park vistor center, not big but big enough to park for awhile. It is just past Game Lodge campground, on the left side of the road.
If time permits, ask if you can unhook and take your truck up 16A (Iron Mountain Road)... it's a beautiful drive.
Leaving the park, continue west to highway 36 and then go north on 79 to Rapid City. Flying J truck stop on 79 about a mile north of junction 36/79.
P.S. I see you want to keep your trailer hooked up. I know that Rushmore has parking for some RVs, but don't know if there is a limit on the size. Call and ask them. That would be your best bet.