You've got a solid outline, and I completely agree with Dave that detouring to Vegas and LA doesn't really make any sense on your timeline.

I will offer a couple small suggestions.

Rapid City is actually quite a bit out of the way as an overnight for what you want to see in the Black Hills, and considering you're already rushing through this section of your trip, you should reduce that as much as possible. I'd strongly recommend trying to stay in Custer or perhaps Hill City. That will eliminate almost 3 hours of extra driving by going from Mt Rushmore, northwest to Rapid, southeast to Custer State Park, and then NW again back through Rapid and onto Wyoming.