We camped in SD two years ago and I drove on the Needles highway Rt 87 and no way can you pull a 5er on that road. I drove my 2008 Mega Cab on that road and had to take the center lane to get through the tunnels with the mirrors folded in. The highway in Custer State park RT 87 is full of sharp hairpin turns and I had to drive this is first and second gear on some of the turns, no way will a 5er travel on this road in the park.
You might be able to go to Custer and enter the park from RT 89 according to their map. Don't know for sure? Might be able to ask a ranger when you get there.
I would also suggest that you try another camp ground if possible. I stayed at Rafter J Bar Ranch in Hill City SD. This is less then a 30 min drive to Custer state park and Crazy Horse. It was a 45 min drive or less then that to Mount Rushmore.
Here is there Web Site if you would like to review it.
http://www.rafterj.com/
Good Luck.
Jim
P.S. MY DW wants full-hookups, Rafter J bar has them.