Hi @Lydia
The father does not have a full right to a residence permit either with or without Sorgerecht. Such a right is only for the parents of German children.
In his case, the authorities have discretion on whether they issue him with a reisdence permit. The relevant type of permit is for humanitarian reasons and is in accordance with paragraph 25, section 5 of the immigration law (Aufenthaltsgesetz). This type of permit is generally given after the person has had a Duldung for 18 months and the reason they have a Duldung is not likely to change. It can, however, be issued before 18 months have passed.
If the father has a Duldung due to your children, then for a residence permit the most important thing is that he has an enduring relationship with the children. Having Sorgerecht can be an indication of this but is not enough by itself and by no means the only way to show this relationship. The authorities must assess whether this relationship is likely to continue based on past behaviour. If the father does not live with the children, then additional proof of the relationship is required, e.g. that they regularly spend time together, going on holidays, doing a significant amount of the childcare (such as bringing to/from Kita and school, going to the doctor when sick), as well as financial support (Unterhalt).
Best,
Éanna