It sounds like you are trying to cut through Yellowstone to save miles. Keep in mind that even though it might be shorter it will take more time - maybe a lot more time. There is a 45mph speed limit that is strictly enforced in Yellowstone. The reason for that strict speed limit is sight lines are often short and animals are likely to run out in the road at any time. I spend a lot of time in the park and have seen many vehicles (motorcycles included) hit bison, elk, and deer because they were in too much of a hurry. Trust me, you do not want to hit a 2000 pound animal on your bike - you will not walk away from it. If you are going to Yellowstone to enjoy the park great. If you are going through Yellowstone and hoping to make time, find a different road. And yes, summertime traffic can be a nightmare.
The western part of the Bear Tooth from the junction with The Chief Joseph Highway is the least scenic part of the road. Chief Joseph to the Beartooth and down to Red Lodge is amazing!!! Go that way and then figure out your route either to I-90 or back roads to hwy 89. I actually prefer the interstate as it is pretty scenic in that part of the world.