You need to take into consideration, that you don't break camp and get ready to leave as you would in a motel, It takes time and even if you don't hook up, we still find we don't leave as early as we might from a hotel.We spent a week in Yellowstone there is a lot to see and do.
As I said, done it twice and it's doable. Our kids are younger and they had the iPad, puzzle books, craft kits, and of course the LCD/DVD system. They would also do some snoozing in the afternoon in their seats. Just be prepared to have a day or two when everyone's had enough being trapped in the MH.
2013 Jayco Eagle 334RBTS Disclaimer for the daft: Don't confuse my opinion with facts.
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.
Maybe you are younger but we never come close to 600 miles nor do we want to. Our goal is no more than 300something, arrive at the campground in plenty of time to get a site and set up for a 6 PM dinner. Of course occassional exceptions are necessary but 450 would be tops. RVing and hurry up quick trips are incompatible in our opinion. I understand your shortness of time but I would reconsider.
I think it all comes down to your ability to drive long hours. I know I can't drive as long as I did when in my 20s. I get really get tired after 350 miles and six hours seems to be my max before I start to get sloppy.
2002 Keystone Cougar 286, 8,400lbs loaded, pulled with a 2004 F150 Supercrew, 5.4, 3.73 gears. Retired and enjoying life
In June 2011, we left DC at 4PM, bound for Glacier National Park. The first night we made it to just past Indianapolis by about 3AM, a little over 600 miles. The next morning, we started around 10AM and made it to just outside Mitchell SD around midnight, a little over 800 miles. The next day we made it to the badlands by noon where we over-nighted, then to Custer State Park where we spent three nights. We had the advantage of no children in the car. We also don't mind driving if we need to, to get where we are going. We also spent 3 nights in Yellowstone, 4 nights in Glacier, 2 nights in Teddy Roosevelt, a night near the Dells in Wisconsin, and two other nights at various spots on the road in Montana. We arrived back in DC on what would have been our 19th night.
It can be done, it really just depends on your tolerance levels. DH and I have very high tolerances and have been traveling like this for years. If we know that we will have a few days down time we can easily crunch out 700 miles. We did it again in September 2012, going 700 miles on day one and 850 miles on day 2. Day 3 we were in Mesa Verde.
Just a heads up though - we can drive these times because we stop at rest stops, not campgrounds. Hooking, unhooking, packing, unpacking - entirely too much time and trouble. But, we don't have kids, don't have to open our slides, and can carry enough water for a shower and coffee. Staying at a campground, you need to arrive by 7PM (courtesy) and be ready to leave again by 6AM just to get enough hours to cover 600-700 miles.