Thank you for sharing your situation with the Wefugee Community!
As far as I can see, you can try to apply for a "Aufenthaltserlaubnis zur Personensorge" as soon as your child has a residence permit. I think you did already good, that you organized the "Vaterschaftsanerkennung" (acknowledge of paternity) and the "Sorgerechtserklärung" (child custody). This will be essential for the application. However, the whole process will be very difficult and there is no guarantee for a success. As you already hired a lawyer, you should ask him/her to advise you in this also (i.e. about your chances in the still pending asylum procedure). At the end, I assume that it would be the best if you receive an independent protection status.
Please do not hesitate to get back to us with any further questions!
All the best, Meike