All that way for 6 days! Nuts! But that is my JMHO having just returned from the mountain states to the East Coast, allowing 5 days for the one way trip.
As pointed out 2 days are lost shuttling to & from airports, never mind a weather day & to & from the only one attraction area that there only just might be time for.
Finding sites within a NP, NFS or SPs for a rig that size is not going to be easy meaning that the camping experience will likely be just as it is at any other commercial CG anywhere in the country & not necessarily close to the attractions you want to see, e.g. staying in West Yellowstone & commuting into the park is a time consuming hike.
Then there is the little matter of fuel burned commuting across the country & what will be the return on all of that expense.
Sorry, but JMO, leave the trip till you have plenty of time.