Wow! Are you aware of the mileage for this trip? It's 1400 miles from Lebanon, MO to W. Yellowstone. Pulling a RV, that's a very long two-day drive counting on no maintenance issues, etc.
The route you would take is I-80 getting off at Rawlins, Wyoming on Hwy 191. Take 191 which will bring you into Grand Teton Nat'l Park. Coming in from this entrance will at least give you a wonderful view of the Tetons since you have no time to stay there.
Then continue north into Yellowstone via 191/287/89 and to West Yellowstone. Skip Gardiner.
Then you only have four nights for Yellowstone and something in Montana. I would suggest going to Ennis/Virginia City for your Montana 'experience'. You just don't have time to get any further and you could do this side trip from W. Yellowstone.
Then you're planning two days to return to Missouri. You are in for some long, long driving days. I hope you have others in the vehicle that can share your driving.
I also see that you have two dogs. I'd suggest you leave them at home in care of someone else for this trip. Dogs are not allowed on the trails of Yellowstone, including boardwalks. Some of the best features of Yellowstone is walking on the boardwalks to get to them.
Here's is Yellowstone's official site with lots of good information. Explore it thoroughly, included the pet regulations.
http://www.nps.gov/yell/planyourvisit/pets.htm