I would not have him at the birth.
Quite apart from his behaviour, if you or DD are unwell and you are unable to decide which treatment you require, the decision will be your ex's. Given his behaviour I would not trust him to make any decisions which are vital to you or DD. He has already shown you how caring he is by leaving you for ow and not bothering to see his son.
I would go in for your c section on it's booked date, take someone you trust as a birth partner. Tell the MW's no one else is to be allowed in, and you can relax and bond with your DD.
After you have left hospital, text PIL to tell them DD has been born, and he can arrange a visit if he wants to. At that point you can discuss him being on the birth certificate, as you will have had time to think about it. I believe you can add him on later, so if you have any doubt about giving him parental responsibility, you do not need to add him when you register the birth.
I can't imagine how you must feel, but I would not let him act as he has and then dare to accuse you of preventing him from being at the birth. He is the one who left you and the children; he doesn't deserve to see dd's birth as he left her before she was even here.
He doesn't get to come back for the good bits.