After figuring out your mileage (11,000 km or 6820 miles) and daily average (180 km, 112 miles), a couple of things spring to mind:
First, add another 25% to the miles for "tourist miles". That's the mileage you'll need for your national park trips, to/from your motel and points of interest within each place, etc. So call your mileage between 8,000 miles and 9,000 miles.
Second, most driving days, you'll find 112 miles to be a very, very short day.
Renting one car for the entire loop is probably the smartest thing you'll do. You should ask about vehicle maintenance while you're on the road. Some rental companies (Enterprise, Alamo, National) may want you to bring the car to the nearest office for maintenance after a certain period of time -- either a specific number of miles or a specific amount of time. Knowing this ahead of time will help you plan where to meet that.