Dear @Borngreat,
Welcome to the Wefugees Community and thank you for reaching out to us.
As far as I know, your child's father could try to apply for a residence permit "zur Ausübung der Personensorge" (to take care of his child). Unfortunately, there is in general no "right" to obtain this type of residence permit. Instead, the issuing is at the discretion of the immigration office. It is good though, that you took already care of the recognition of the paternity and the declaration of the shared custody - these steps are essential for a respective application. Please see more exchange on this topic here or here, for instance.
I will link two of our experts here as well. Dear @mbeon-fardeen or @mbeon-Ruth, do you agree with my explanations and/or is there anything you would like to add?
In any case, please don't hesitate to get back to us with any further questions.
All the best,
Meike