Two thoughts are going through my head. This isn't directed to anyone in particular, but why do some think just because a family is larger than average they should be exempt from rules such as max numbers on a campsite? My concern would be some nosy nellie doing a head count and filing a complaint or some campgrounds making the max number even smaller as a result.
Second is I always camped with a max of three, but always paid the minimum rate which usually covered 4 persons. Should I also be exempt from the rules and pay less? I never did-I just decided it was a cost of being an RVer. Rules are based on averages and those of use who fall on the outside of the bell curve shouldn't expect special treatment. I fully understand the OP isn't expecting such treatment.
OP I just did a Google search on "campgrounds for large families" and it popped up a bunch of local campgrounds plus Tripadvisor. Try searching that way and see what happens. Group areas are always a possibility but lack of hook-ups may be an issue for you.