Get a full size body on frame VAN.
2 Bathrooms: Unless you plan to boondock a lot, plan on using the campground bathhouse unless it's a late night need. Even boondocking, water consumption and waste storage will be very difficult to last more than a night or two with good water conservation.
Keep in mind, those kids will want to bring along friends...or they won't want to go camping with you.
Payload: Just bodies could eat up 800-1000lb of payload before putting anything in the truck or the trailer on the hitch.
PS: Youngest of 10 here, took 3-4 week trip each summer...