Thank you for your question, and welcome to the Wefugees Community!
With the Vaterschaftsanerkennung and the Sorgerecht for your child, you should be able to stay in Germany. @Nilab answered a similar question a few months ago. Have a look at it to learn more about the process to get the Vaterschaftsanerkennung, Sorgerecht and apply for a Aufenthaltserlaubnis.
Since your asylum claim has been rejected, the Ausländerbehörde might investigate to make sure you don't pretend to be the father in order to stay in Germany (§85a AufenthG). They might ask for a DNA test, for example. The process until you receive your Aufenthaltserlaubnis, which is already quite long, could take even longer. However, according to §60a Abs 2 Satz.13 AufenthG, you have the right to have your Duldung extended for the entire time of the procedure, which will protect you against deportation.
I highly recommend you to get legal support in order to make sure your rights are respected. In Munich, you can go to the Rechtshilfe für AusländerInnen every Tuedsay at 17:30. They have lawyers counselling persons for free.
I hope this answer helps you, don't hesitate to add a comment if you need additional information!
All the best,