Will you have seen Yellowstone by that time?
Where are you going after Cody?
If you are in GT and haven't seen Yellowstone yet, the do that as part of the drive to Cody. Also if after Cody you intend to head south the I80, you might as well skip the 2 way drive past Thermopolis (not that there's anything wrong with that).
But if you've already seen the Park, and will be heading east over the Big Horns, then the southern route would be fine.
Years ago, I drove Chicago to Seattle via this southern route. I'd already done Yellowstone and Cody on a previous trip. So I took US16 over the Big Horns, then straight to GT, and on west into Idaho.