Hi @Lloyds
Before it was rejected, the Jugendamt must have given you a hearing (in German ‘Anhörung’). This means they would have told you they planned to reject it and why, and will have given you the opportunity to respond to it. If they still believe it to be a misuse after the hearing, they then pass it on to the Ausländerbehörde who check the situation further.
It can be rejected for specific reasons, such as when a person who is obliged to leave Germany tries to do it. Or if the father has already done several Vaterschaftanerkennung with different mothers.
This does not apply in your case as you have an Aufenthaltsgestattung and are not obliged to leave Germany. It only applies to asylum seekers if they are from what is called a safe country.
It is difficult for me to recommend what to do without knowing whether there was a hearing as part of the process and the reasons for the rejection. It is possible to do a DNA test to provide additional proof. However, this can be expensive and so it is first important to find out why exactly it was rejected and if this decision was made correctly.
Best,
Éanna