As a matter of fact, we drove it today in our Sprinter with room to spare. Tunnel Five (Needles Eye) is the tightest at 8'4" wide. We had a lot of fun with it, jumping out to take photos like many folks do at the tunnel exits. An incredibly scenic road.
Started the day at 7:15 a.m. on Iron Mountain Road from Game Lodge CG in Custer SP to Rushmore. Then Needles. Late today around sunset, drove the Wildlife Loop. Dozens of buffalo crossing the road. What a great destination. South Dakota has greatly exceeded my expectations.
But no, not all of the six tunnels have a bypass, only two do. These roads are narrow, seldom have guard rails, and are full of tight corkscrew turns. Fun!