How long do these face time calls usually last? I can see that they would be nice for him but are they of any use to the baby? I really can’t see a judge ordering anything more than twice a week to be honest.
I would say he is being massively premature in threatening court. Most likely, it is just a continuation of his manipulative and controlling behaviour.
The fact that his mother went to court and understands the system for all of her 8 children with 3 fathers (so potentially only 3 court cases) is irrelevant. Her situation is different to yours. If that really happened, she would know and advise him that contact with babies is advised to be little and often with no overnights until the age of 2+.
So...put it in writing - I am not withholding contact. You said you couldn’t make the call as planned today and I accepted that and said ‘ok, next week then’. Do you have an alternative? (You are demonstrating being open, reasonable and flexible).
If it is convenient for you, accept his alternative. If not, suggest something else. (Remember that he cancelled contact first, not you).
Is he saying you are withholding contact because he hasn’t seen her in person? If so, I can understand his upset. Unless you have very specific medical advice because of COVID, make arrangements to let him see the baby little and often. You can say that because of covid it cannot be in your home and meet in the park instead. Given the baby’s feeding schedule, the visits will necessarily be short because you may be more comfortable returning home to breast feed.
Alternatively, you can dig your heels in and refuse to have any further contact. If you go down that road, change your phone number and don’t do anything until you hear from his solicitor/the court. Communicate only through legal channels. It will cost him money and he may not be bothered enough to try. If he does, he will likely be given access but, again, little and often is recommended for babies and overnights won’t happen for a couple of years.