Thinking about going to Mt. Rushmore, Crazy Horse, etc. in Sept. Usually we like to reserve hotels in advance but this time I'd prefer to have the flexibility of not doing that. I'm guessing Rapid City itself wouldn't be an issue but have no idea about smaller places like Keystone or Hill City. Any thoughts about how busy it would be at that time of year? Thank you!

You shouldn't have to worry in September after at least most of the kids are back in school. The Mt. Rushmore Lodge at Palmer Gulch looks nice. They also have some 1 and 2 bedroom cabins with kitchenette in addition to the lodge . Check if you need reservations about the second week of August for your favored lodging.
While in the area, check out Wind Cave NP which is often ignored.

When in September? I would expect Labor Day weekend will be busy and possibly the following weekend. If you're midweek later in the month you'll be fine.

It would be mid month. I've been looking at different places to stay and many of them already are booked up.