Checking new route from Black Hills to Cheyenne, WY

Or you can take I-90 west into Wyoming.

Then go south off I-90 at Sundance, Wyoming and take Wyoming route 585 until it meets up with US 85 NE of Newcastle, Wyoming.

I've taken my 35 foot fiver through this route, for same reasons you give (avoiding mountainous roads) and found WY 585 to be just fine.