this document explains the procedure for obtaining the birth certificate for your child very well: https://www.institut-fuer-menschenrechte.de/fileadmin/user_upload/Publikationen/Weitere_Publikationen/Info_How_to_Register_Your_Newborn.pdf
Unfortunately, it happens quite often, that the local civil registry office ("Standesamt") is not accepting parents' birth certificates from foreign countries if they doubt the authenticity. However, you should always be able to receive an "extract from the birth registry" for your child. As stated in the brochure, the extract is provisional, but an official document and not less so than a birth certificate.
I hope this information will be useful for you and please get back to us if clarification is needed.
All the best,