I don't understand why some of the folks on the forum think a 30 ft TT and toad is a "large rig". We have stayed in YNP with 34ft 5er and 1 ton dodge dually and did not have any problems.
I would probably go ahead and get reservation at Madison or one of the CGs for a couple of days and visit the areas near there while scoping out what part we wanted to see next then get reservations at another area.
DO NOT try to stay outside the park as it is a Long, long drive into the park to whatever area you are seeing that day. Some people will tell you that the Northwest entrance from West Yellowstone is just a short distance and they are correct, but that is just the Entrance. It is not close to the attractions.
I would not pull the trailer somewhere and leave the wife and kids somewhere else. You have no idea how big YNP is and it would take you at least half a day to do that. What a waste of great sightseeing time.
Anyway, have a great time!
