Welcome back to the Wefugees Community and thank you for reaching out to us again.
I am really sorry to hear that you are struggling so much to get the birth certificate or at least the extract from the birth registry (Auszug aus dem Geburtenregister). From my experience, the Standesamt is usually issuing the latter BECAUSE the parents' identity could not be proven. I recommend you to address the Standesamt again and request the immediate issuing of the extract (Auszug aus dem Geburtenregister) while highlighting that otherwise your child is not even receiving any benefits, medical services etc. Explain them that you are not able to submit valid passports for now and you REQUEST the finalization of the procedure. Until now, they could have issued a so-called "Bescheinigung über die Zurückstellung der Beurkundung" at least (which they obviously didn't do either). Please see more details on this here.
The other question is, what is the lawyer actually doing? You hired him/her for this matter? With this kind of issues I would recommend to contact a counselling office first - it may save the money for a lawyer.
Let us know if you need support in finding counselling services in your area and I hope my answer was helpful.
All the best,