Traffic in Yellowstone and the Tetons can be a bit daunting, hard to grab a space, etc. etc.
When I was out there (over a decade ago, mind you), what we did was: coming up from the south (through Pinedale), just below Hoback Jctn (a little bit south of Jackson), was on Granite Crk. Beautiful spot. We camped there for a couple of days and day-tripped into Tetons. Then, we packed up, and went to the north side of West Yellowstone and camped just south of Big Sky, and day-tripped into Yellowstone for a few days.
I don't think having dogs with you would be a problem in nat. forest, but it might in a nat. park. So, camping outside of the parks proper might be a good idea....
FWIW...