There is good climbing there. I don't know of any offroad trails personally, but since it's surrounded by the Black Hills Forest, there has got to be something, especially fire service roads. I can recommend you guys stop in Keystone for some lunch. And you should check out Sylvan Lake in East Custer, SD about 20 mins from there. I haven't been there since 2009 though.
Also the needles is a climbing area real close to Rushmoore that you can drive thru and has some crazy narrow one way sections with collapsed over head rocks. Worth seeing!
Once lifted 03 Trailblazer on 35's, gave it up to travel the USA with my family.
There are numerous trails in the Black Hills near Mt. Rushmore. There is a local 4X4 club in Rapid City you should contact regarding what will be doable. http://www.bh4wheelers.com/ You missed their DTC for the year, but I'm sure you'll find someone on their forums willing to give you some information on where you can go.