It's doable but more of a road trip vs get someplace and camp. Plugging into google maps, it's about 800 miles, so figure around 16hr of driving (you go slower in an RV so we usually figure a 50mph average speed). Add a half hour on ether end for set up and tear down. Over 7 days it's doable.
What will you be using to tour around in? It's a pain to pack up the camper every time you want to go sight seeing or even to the grocery store, plus in many areas parking for a motorhome will be tough. Most owners, tow a car or go very small on the MH...not real practical flying out with larger group.
I would step back and ask what are you trying to see and how much driving do you want to do.
What is your experience driving large vehicles? Nothing horrible on the route but some descent grades. Presumably a rental will be in good operating condition but do a little research on how to drive on steep grades.
Have a little discussion about bathroom usage with the family. You have limited fresh water storage and more important gray/black water storage...if possible use shower house or other facilities when available.
Keep in mind most rental units are minimally equipped, so boondocking (camping away from campgrounds with no electricity or water) requires you to be careful with usage. Also, you probably won't have a grill and cooking equipment will be minimal, so how are you feeding everyone...speaking of which, a big chunk of the first day will be lost getting orientated and then hitting a grocery store (assuming you won't be doing all restaurants).
None of this is intended to turn you off to the idea it's just easier when you own the rig and can outfit it the way you want.