I drove the reverse of your request last week towing my Montana. Not the shortest as Glenn provided, but certainly scenic. Here is the path I went (for your perspective):
From Rapid City I90 West to exit 58 (several large fuel stops off exit 58). Follow 16 West across Big Horn National Forest - we stayed in the forest at Boulder Park, a new national forest CG (no facilities, but very large back in sites). Big Horn has a couple of grades, but nothing that worries. We saw several moose while traveling through the pass. Stay on 16 through Worland (several fuel stops & grocery stores) and Greybull (both had many campgrounds). At Greybull take 14 West to Cody.
Take care,
Joe
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