From what I have heard, traffic after July 4 is much worse than before July 4. My experience on the TransAm, as I would guess most cyclotourists' experience on the TA, is of Yellowstone before July 4, and I didn't find the traffic to be a problem at all. One of the nicest things about being on a bicycle is that you can cruise past any traffic obstructions caused by animals on the road or animal sightings off the road. I cruised by a 5-mile backup of cars trying to enter from West Yellowstone because of bison on the road near Madison. Since I was going in the other direction, I had my side of the road all to myself.
There aren't many roads in Yellowstone, so planning a route is pretty trivial. The main roads form a figure 8, so you either do the upper loop, the lower loop, or both if you have time. Campgrounds in Yellowstone have hiker/biker spots at good prices, so that's not a worry.