As already mentioned, during late August there will massive crowds in the park and yes, people do stop in the middle of the road backing up traffic for such common stuff such as a chipmunk. OK, that might be slightly exaggerated (not really) but the point is if you can travel through the park early in the morning, you will get through it much quicker, will be a lot less lookyloo's.
I towed my 30' TT with a crewcab/longbed truck from West Yellowstone to Cody and there were no issues doing it. I'd do it again and would not consider going a different route.