My wife's family is from Custer, so we have been there many times over the past 25 years. On the Needles Highway, there are three significant tunnels. Like the picture above, I have one of those pictures on my camera of a tour bus going through the smallest of the 3 tunnels. I was in our van at the time, but being right behind the bus, I was interested because I wanted to know if an RV could fit through. The bus had a much tighter fit than the photo above.
The professional tour bus driver took about 10 minutes to go through a tunnel less than 200 feet long. We followed the bus to Sylvan Lake. The bus stopped there and so I decided to pull up next to the bus. There where scrape marks all down the length of the bus. It had to be a 45' motor coach and no telling how much damage there was.
While I highly recommend touring the Black Hills, I strongly suggest you carefully measure your RV height AND width. If there is any doubt, chose another route. I have 36' National RV Dophin and I will not attempt a tunnel.
Needles Highway is beautiful, but as an alternative, take the Wildlife Loop early in the morning. More than one I have stopped for a herd of several hundred buffalo crossing the road. If you go in the middle of the day, you are less likely to see a lot of wildlife.
Have a great trip.
