Hi,
I left the Philippines in June with my Fil wife and our daughter who also has a Brit and a Phil passport. The correct thing to do is to show BOTH of your childs passports every single time you are facing any immigration officer no matter where you are. And they will then do what is required. There will be Fil travel tax and terminal fee to be paid at Manila airport before they can fly to join you. But its just a matter of going to the desk and paying the pesos.
