We will be going through there in early September, although we have a much smaller unit. September should be okay weather wise, but it will be cool to cold at night. You should expect light traffic as the summer travellers will all be home with kids back to school.
I-90 west out of Rapid City is a good road. How are you getting to Rapid City? If you are coming across I-90 from the east, a word of caution as there is major highway construction on I-90 across eastern South Dakota. Expect some delays and lane closures. With a little luck the construction will be mostly completed by the time you (and us!) go through there.
I am not sure of your best route once you get out of Rapid City and to Buffalo, WY. Do you want to go through Yellowstone? There are mountain passes west of Buffalo on highways 16 & 14. You may want to take 87 south to Capser and then proceed west.
That the best help I can be. I have been on most of those roads, but on a motorcycle so I was not concerned about mountain passes or tight corners.
Hopefully someone here on the board has been through that area in a big rig and can provide the information you need.
Enjoy you trip!