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. For what it is worth we rolled through after the 4th of July and it was still OK IMO.
We were a bit over a week (and a year) later and it still wasn't too bad. Worst part (coming from the south) was the long lines coming through the road construction (since finished) and trying to make up for lost time past West Thumb, and we didn't have much problem there.
As Pete says, do plan to get off and hike a bit. It might be worth a half-day tour on one of the buses to see part of the thermal features -- get out and hike up to some of the pools and geysers. After about 4-5 hours, they start to look alike.
Oh, and FWIW, the south and west that the TransAm covers are about the worst parts of the park Go north and east for the best scenery (and most of the trees!).