Your wife should be able to get a birth certificate (even if 'late registration') locally.

Just make sure YOUR NAME is on it as the father.

Births are often delivered by a local amateur midwife in the province, and the Phil govt. recognises this...there is a word for it, that I forget.

Anyway, once your wife has the local certificate it has to be be sent to Manila to be registered at the national statistics office.
THIS document will then be used by our Embassy.

I went through all this 'late-registered' provincial birth stuff when I brought my stepson over.

Some people don't register births until the kids are several years old !

Sounds like you have all the other docs.

Good luck.