Hes no longer happy with seeing her with me and wants to take her every other weekend and then the weekend he has his boys a Saturday day or Sunday day so they can bond.
If he wants to see her, he should be seeing her 3-4 times per week for about 45 mins to an hour with you there.
Then once she recognises him at about 5-6 months he can do the same on his own for a couple of months. In other words little and often.
Then he can start having her for hours at a time in the day.
Only after she knows who he is and they have an established relationship, so after a year, can he start having her overnights.
I know fathers who have and have not gone through the Family Court to establish relationships with their babies.
The ones who put in Court applications, so got initial hearings, got interim orders until the baby was over a year. They actually started their overnights at an older age than those who hadn't gone to Court.
Basically put your foot down with him about the overnights and stick to what you are doing now.
If he doesn't accept it then tell him by email that the contact what he's suggesting is not in your joint 6 week old baby's best interests.
Do not say anything else.
Leave it to him to take you to mediation and court.