[twindman]

I have driven thru Yellowstone from Cody to West Yellowstone probably 8-10 times in the last 16 years - pulling my Montana. I am not worried about mountainous driving tho. I don't think this road is treacherous or bad at all. Typical hills, but roads are moderately wide and not super steep or curvy.

Yes, thru Fishing Bridge it is busy, but most of the rest of the direct route isn't - well, except animal sitings for 20 miles east of W. Yellowstone. That varies quite a bit from some stop and go to just normal 30-45 mph.
So take your time and stop at Fishing Bridge for lunch. There is a rest area (?) across the street from the General Store where there is a dining area.
It has been a while but one time we spotted a 1 or 2 year old grizzly kind of wandering around the intersection west of the river. People said it was kind of lost and confused and had been spotted for several days.
Good luck.
Oh, I have also take the route on the east to I90 down to Badge, Belfry to Ralston. But you can cut over near the southern end and go straight in to Cody. I think these roads are pretty wide open and mostly level.