If Yellowstone is your only destination you don't need to stay 10 days. Five would be more than enough. With the extra time, stop somewhere after two heavy driving days and relax for two days before carrying on.
Only you know how well your children would do in this situation but three larger kids sitting close together and unable to stretch out for 12+ hours would be torture for them...and probably for you.
Your trip can be done...many folks only have two weeks vacation and do it. We did...from Michigan with the kids - tenting. However, we broke it up in more relaxing chunks of time.
Good luck. Hope it works for you.
