Reminding him to care for his kids

by treating him like a kid will not help. Yes, making clear that you expect him to share the care work with you is important. Making rules together can be a way of doing it, but he needs to do it because he is the dad and her partner and a reasonable adult that takes their responsibilities serious, not because he wants a BJ at the end of the week. They both need couple therapy, because he isn't a responsible adult and she infantilizes him on top.