From Rawlins Wy to Mt Rushmore

Hwys 287 and 34 out of Laramie should be fine. 287 is straight and flat; 34 drops down into Sybille Canyon and is a bit hilly and curvy but nothing to avoid in the summer. (It can get nasty with winter storms.) You might want to stop at the Sybille Wildlife Research Center for a short break if you spot it. It's not big, but it's handy right along the highway. I've stopped there a few times to get some easy wildlife photos. The only thing is, I can't recall if there's any parking for a rig. My trips through there have always been in a car or pickup, usually heading to/from Laramie while visiting my kids at UW, attending a football game, etc.

I wouldn't have thought to take that route, but since you mentioned it, it sounds good to me. The more typical route would be I-80 to Cheyenne, then Hwy 85 north.