My wife and I did a road trip out west this past summer (from CT). Took just under 6 weeks (recently retired) with about 8,000 miles of driving overall. I brought my bike as I had a couple of bucket list rides I wanted to do although it wasn't primarily a cycling trip as my wife doesn't cycle. Did more hiking than cycling.

Having said that, our trip was further north than the options you noted although we did overlap with Colorado. Our first major stop was Colorado and we did the auto drive up Mount Evans (I didn't try cycling that one as I wasn't anywhere near acclimated to the altitude yet. This was that latter part of mid-June and on the day we drove, the upper section past Summit Lake was closed off due to snowy/icy conditions. That's just to confirm what others have said about any higher altitude climbs you may be planning. Some of the highest can get snow any time but you may want to go later than May - maybe even into July/August to increase your odds.

The two bucket list rides I did were Going to The Sun Road and Beartooth Highway. Spectacular rides and in the ballpark (maybe less) of the amount of driving as the other options you noted. On the way back, we stopped at the Badlands National Park. I hadn't really read a lot about cycling in the Badlands - it wouldn't show up in hillclimb sites like PJAMM as it's more rolling than big hills - but I had just a wonderful, beautiful relaxing ride one early morning before any traffic came out. Really a fun ride.