Later is better, I've motorcycled through the area multiple times. Figure out a way to get to Devils Tower in NE Wyoming. It's a little out of the way but well worth the trip.
My opinion, stay off the interstate and take US14 as much as possible, look at this route from Devils Tower to Yellowstone east entrance.
From Devils Tower, take US14 to Moorcroft then 90 to Gillette then 14 to Sheridan.
I-90 to Ranchester the US14 to Burgess Junction. You have a big decision to take 14Alt to Lovell or 14 to Greybull, both are fantastic drives with great scenery and views. If it were me, I'd take 14 to Greybull and a few miles past, take 310 to Lovell then 14Alt to Cody.
From Cody, 120 north to 296 then 296 over Chief Joseph pass to 212 over Beartooth Pass to Redlodge. Beartooth can be closed until July in years with a lot of snow.
Then backtrack back over 212 to Cooke City and Tower Junction in the park. At Tower Junction head south to Yellowstone lake.
You're on your own from there.
What were your plans for getting back home?
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