I don’t want to frighten you more than you already are, but it could get a lot scarier unless they do something, fast!! I’m going to deviate slightly from your question about the father’s name on the certificate, because, reading your post, I believe the situation is urgent.
The most important piece of advice is GET THE POLICE INVOLVED ASAP. Because there has been violence and threats it is essential that the police are told. A restraining order can be applied for (but not necessarily granted) if the police think that the threat is great enough then a panic alarm could be fitted in your home in case he tries anything there. My advice is if they feel the threat that seriously she should NOT leave her daughter alone, ever, until it is sorted, that way she always has a witness, and protection. If this is impractical tell her to carry a personal alarm and digital recorder with her everywhere and if he comes near her, use them! If she has to go anywhere on her own it is vital she tells the people/place she is leaving where her destination is, it is also vital at the same time to inform her destination where she is coming from and her estimated time of arrival, even for something like going to and from work, that way if anything happens the police can be informed immediately and something drastic could be avoided.
If you can afford it speak to a reputable family law solicitor and get professional advice. If not the Citizens Advice Bureau is your best bet, if they can’t handle it they’ll pass you on to someone who can. Legally get witness statements (before he can intimidate any of them) to any previous abuse; if anything goes to court it could come in useful.
Regarding access, if she once mentions he is the father she’s done for in my opinion, please note that this is MY opinion only. Social services and courts think the sun shines out of the father’s backside even if he is violent or showing threatening behaviour to either child or mother. To make you understand how serious that this situation could be, the worst case scenario, so to speak, my family tragedy ended in 2 children being murdered by their father, he had threatened to kill their mother and them, but social services and the courts said he had a right to see his children un-supervised. During his first access visit, he took their lives. THIS IS RARE but it does happen. I am sorry to frighten you, but as a family, we didn’t believe this could happen either, but it did.
She is legally within her rights not to name the father, though if they are unmarried and he wants to be named as the father on the birth certificate then he has to attend with the mother at registration, so unless some sly ‘friend’ tells him when and where your friends daughter is registering the birth or he rings the registration office and enquires when the appointment is, she should be able to do it alone and leave his name off the certificate. As I understand it he has the right, if he wants to, to apply for DNA testing, it’s not cheap, and as the applicant I believe he has to pay. Not sure what happens after the tests in this type of situation.
I hope that this situation turns out to be not as serious as your post sounds and my advice is not needed, but if this helps one person to keep themselves safe it was worth the typing. I wish your friend and her daughter much happiness with their new addition to the family when he/she arrives.