Another vote for Grand Teton NP, as well as Gros Ventre campground. We have visited at least a dozen times since the 70's and always stay at Gros Ventre. One loop with electric and water available at he dump station. X2 on 2gypsies campground suggestions for Yellowstone. Forget the suggestion for Lewis lake campground - 25ft length limit.
Leave the bikes at home. There is a paved bike trail between the town of Jackson as far as Jenny Lake, but you can make better use of your short time available.
If you are interested, take the TV down to Schwabaker Landing and hike the short trail to the beaver pond. Another easy flat trail goes from the String lake picnic area all the way to Leigh Lake. The boat ride across Jenny Lake and hike to Hidden Falls and up to Inspiration point is more strenuous, but worthwhile.
