Bridge Bay and Fishing Bridge are only a few miles apart, so IMO it would be a waste of time to move between them.
Most of the thermal areas are between Madison and Old Faithful, so a few days at Madison would save quite a bit of driving. If it were me, I would try to get reservations at Madison and Canyon instead of Bridge Bay. Canyon is obviously good for the Yellowstone Canyon as well as down the east side of the lower loop to Fishing Bridge and would also be better base for doing the upper loop.
There is a
KOA a few miles east Cody on route 14. Closer in, but still out of town is the
Absorka Bay RV Park. In town a couple of blocks from the Buffalo Bill Historical Center is the
Ponderosa Campground. Another possibility is the
Buffalo Bill State Park, about 7 miles west of town. Call them for info on lead time for reservations for the 4th.
Suggest you fill up with gas in Cody and stop at the WalMart on the west side to restock, as both are much more expensive inside Yellowstone.
In the Tetons, there are both scenic and white water raft rides out of the town of Jackson. There is also a nightly shoot out at the Jackson town square and a rodeo on Wednesday and Saturday nights.
The most popular hike in the Tetons is to take the boat ride across Jenny Lake and hike up to Hidden Falls and on to Inspiration Point for great views across the valley. From there you can continue on into Cascade Canyon to really get into the mountains with not much more uphill effort. Lots of other hikes into the mountains on south from Jenny Lake.
I know there are horse rides in Jackson Hole, but no details. Try a Google search.
Nothing wrong with Colter Bay, and we have stayed there a few times, but our favorite for the Tetons is Gros Ventre on the southern end of the Park. IMO closer to the action.