Yes, I did basically that in the other direction as part of a cross country trip. I would highly recommend it. See
link for google map (ignore eastern portion), there were a few deviations off the 14 (I wanted to go to Devil's Tower, see the badlands, etc).
I've crossed the Bighorns twice, once as you see on the map, west to east, via Worland / Tensleep / Buffalo and once east to west via Ranchester / Granite Pass / Shell Cyn / Greybull. Both were excellent, I would repeat either, but preferred the southern route b/c it was more rugged and remote and higher. The northern Lovell route is purported to be very very steep (uphill west to east, so downhill for you).
The segment between Gillette and Buffalo is quite empty (hilly, hot in summer, windy) but there are two tiny towns, Spotted Horse (population 2, I'm not kidding) which is just a bar in the middle of nowhere, and Clearmont which has a bar and a gas station and a park. It's just enough, but be aware and carry extra water.
Wyoming was awesome.
By the way, when entering Yellowstone, the road between Cody and the park to the west is ok, but if you can manage 120 north to 296 (Sunlight Basin Rd) to Cooke City, that is one of the most spectacular (and remote, and hilly) roads I have ever been on. Hard hills, but incredible riding - no cars, incredible scenery. I also hear the Beartooth Hwy (east from Cooke City) is great, but didn't make it there (yet).
Have a wonderful trip.