3 kids, a wife and an RV for some of it? If you’ve got enough arms to take a bike you’re more octopus than I.
Rent if you feel the need / have the time. You’re looking at a LOT of driving, especially with 3 kids across a wide spread of age.
If you go to Moab, Magpie Adventures do some great guiding and have local knowledge beyond compare.
I’ve driven around that part of the world and you are talking a lot of driving time, especially with kids to entertain as well. I’d try to plan the route to drive less and spend longer at nice places.