Good luck OP. It's funny, hotels in North America generally don't seem to care about how many kids you squeeze into a room, but Europe, Asia, Australia... they all seem follow the letter of the law when it comes to their room capacity policies.
As others have mentioned, airbnb (or vrbo, or any other private rental) is a great solution to this problem. It has been a godsend for me personally. Going airbnb has certain other perks like having a place to prepare/eat breakfast (which is a huge timesaver for us in the morning), and the fact that you can have a separate bedroom to put the kids down to sleep in - it makes life easier when you stay up a bit later.