Birthcertifican

I'm not 100% about time restrictions with immigration butvif indonesian birth certificate I am not 100% why you need to we haven't with our son, but an old school of thought is (again it may have changed) if the child is born here in Indonesia it is usually classed as Indonesian and children up to the age of 18 (could be 21?) can have dual nationality, this could change in the future but for time being only children upto 18 possibly 21 can do this (& Indonesian professionals abroad etc)
With regard to having the fathers name added to the birth certificate, the simplest and easiest way is to marry the mother if child staying in Indonesia or will eventually live in Indonesia it will make the full process simple for schools etc firstly you marry, if mixed faith this will have to be done outside of Indonesia (im getting married Thailand in 4 weeks (I have no religious faith) then you get a DNA test the and get an agent to assist or re-register the birth remember to take the old one and details of the birth
2nd way but is very difficult you can get a DNA test and you need a family lawyer, local court decides if they will allow this, they is no set rule and totally the judges discretion for example if you went to court and said we will marry one day the judge will almost certainly deny your application and tell you to wait until married, if you tell the judge you won't be getting married he could well reply you should get married (single mums is very much taboo in many parts of Indonesia) if you have an Indonesian birth certificate the child can usually get a passport within 2 weeks (can only speak for batam we paid 1.5 million ( or 1.3?) for agent this included the passport)  with the registered parent not being Indonesian this could well be difficult but I can't be certain as other documents are needed and could pose problems taking the child out of Indonesia, maybe get in touch with your Embassay I'm sure they'll help and assist,