Dear @Faga0206,
Welcome back to the Wefugees Community and I am sorry to hear that you have to suffer so much in the German bureaucratic jungle.
You are absolutely right: As long as you are in the asylum procedure (and also as a recognized refugee) you should not ask for services from the embassy of your country of origin.
Unfortunately, the civil servants at the registry office ("Standesamt") can decide quite independently. Therefore, people make very different experiences in the process of the birth certificate application. If the responsible civil servant is deciding now that he/she can not issue a "proper" birth certificate (even though I really don't understand why they don't register the child's father in the birth certificate as you submitted the marriage certificate indluding the legalization etc.), they will issue (and have to) a so-called "Auszug aus dem Geburtenregister". This document is also accepted by the German authorities.
I am not a lawyer, but from my experiences (with similar cases in Berlin), people (esp. with all these documents you also hold) had really good chances at the family court to eventually get the birth certificate ("Antrag auf Berichtigung des Geburteneintrags"). As far as I know, it is also not necessary to hire a lawyer for the process. Maybe you can contact a local migration counselling office to double-check.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,
Meike