Can't tell from your post if you are doing a credit card tour, car/hotel based day rides or self-supported/loaded tour, so it's a little hard to make a recommendation. I'll just point out a few roads that are good, and leave the logistics to you.
That area is gorgeous.
At the Eastern edge of Yellowstone, I really enjoyed Cody To Cooke City via the Chief Joseph Hwy/Sunlight Basin Road (120 to 296). This is a very remote & un-travelled road with spectacular scenery. There are almost no services on it, though, so definitely be prepared with food & water.
I've heard from reliable sources that the Beartooth Higway (212) from Cooke CIty to Red Lodge is absolutely spectacular, but haven't done it myself yet.
Closer to Jackson, you could head out towards Grand Teton Nat'l Park and head up Togwotee Pass, for a big out-and-back climb. They were re-paving it in 2008, so I bet it's awesome. The views coming back towards the Tetons are spectacular.
In August the tourist traffic in Yellowstone itself could be a little too heavy for the Yellowstone roads to be that fun. Also check with the park for bison mating season, I think it's around then - can be scary.
I'd suggest posting on
www.bikeforums.net/forumdisplay.php?47-Touring touring forum and hoping to get a reply from user Jamawani, who lives in Buffalo and knows Wyoming roads really well.
Sunlight Basin Road
