I fully sympathise, as I was put in a similar position with my (ex) wife and son, the difference being that she was in the UK and I was worried about her taking him out of the country.
I got a court order from a Judge preventing her from doing so.
Anyway, regarding your situation, and looking at things totally pragmatically, your child is a British citizen so in theory the Phils government have no jurisdiction over him. That's the way I see it anyway, but you may wish to seek legal advice about that.
With a BRITISH passport or travel document, and being the age he is, there should be no reason why you cannot fly out of the country with him.
There should also be no reason for you to explain the purpose of your journey to any authority. You are his legal guardian/parent, and the British Embassy will only need the supporting documents to enable replacement of his passport. (Birth certificate ?....and preferably copy or at least the number of his original passport.)
