You are talking about 4 12hour days driving one way. With 18 days I would suggest you drive 2 days, stop a day somewhere and then drive 2 days again. Do the samething coming back on a different route. Leaves you 8 days in yellowstone.
My family spends a lot of time on the road showing horses during the year. Most drives are 300 or so miles but about 4 trips are 600 miles. We do the 600 in 1 day but I can tell you nobody is happy at the end of the one of those days. Now keep in mind we are pulling big heavy trailers (2 18k# horse trailers, a FW and an equipment trailer pluse other families occassional with TTs and FWs) I imagine a Class C is probably easier on the attitude than those monsters with live animals aboard.