Change child surname to her father surname

Good Day.

seeking advice regarding my daughter surname in her birth certificate.
My daughter was born June 2010 and i registered her Feb 2011 (late registration), under my name with UNKNOWN father, although her father and i was married year 2008.
Hindi ko po isinunod yung name niya sa father niya kase po were not actually legally married in his country (his saudi and need approval from there government to marry other nationalities), pero sa philippines po were legally married. plan po kc namen sia ilabas ng bansa under my name kaya sa pangalan ko sia isinunod.

We got the marriage approval last nov 2012 and now im living with my husband legally here in saudi arabia.

Ngaun po ang problema ko is pano ko isusunod ung name ng daughter ko sa father nia? Ano po ba easiest way without filling petition in the court. Ang kelangan lang kase po sa Saudi embassy dyan sa Pinas is her birth certificate under her fathers name para ma process ang visa and citizenship nia. (the embassy just need a proof that she was born this date and under her fathers name).

hoping for your advice regarding my problem and thank you in advance.

na try mo na bang ipaasikaso sa father ng bata? pwede naman nyang mapagawa ng affidavit of ackowledgement yun eh! tapos ipa authenticate mo sa Malacanang tapos dalhin mo sa NSO at dalhin mo na rin yung marriage certificate na naaprovan ng Saudi Government para may update ang information sila sa NSO ng validity at acknowledgement ng Saudi government sa marriage certificate nyo saka kayo pumunta ng Saudi Embassy para mabigyan na sya ng passport!

actually po nag inquire na ako sa municipality kung saan ko sia niregister. They said there's complication bec we were married before she was born. so either file a petition for changed of name or just submit acknowledgement letter of the father so no need court petition. But she will be illigitimate child according to them. bec in the birth cert it will not be mentioned that were married.

can it be possible to do it that way, the acknowledgement of paternity from the father only, so no need for court petition which will take long.
The embassy only wants a written proof that my daughter was under her saudi father.
or there's other or best way to transfer her name under her fathers name?

were plan to go vacation to process her papers in civil registrar, i just need advises wat to do, so we dont need to file petition in court.

thanks

see private message!