I can see three main points here.
The first is that, as many posters seem to agree, birth certificates are an important legal document that form a record of who a child's parents are. Everybody should have access to that information about themselves. Given that people seem to agree on that, it is quite extraordinary that we ever allow that to be threatened by using it as a mechanism to award fathers parental responsibility. As ten percent of children are born to non resident fathers, many of who either are not suitable to have or do not want parental responsibility, why are they awarded that through the birth certificate? It must put women off declaring a father on the certificate.
In your case Jackie, the father is resident and has been for a long time, so he would easily get parental responsibility. For any practical purpose, you are not any more secure by keeping his name off it. It is not going to change anything about what could happen to you in the future legally if he is named.
So the real issue is that a. you both came to an agreement over something that was extremely important to you, b. you became pregnant believing that agreement would remain and c. now that you are pregnant, he is going back on that agreement.
Some of the people on this thread believe that asking someone to do that makes you an abuser. If that is the case, then surely they should be advising you to leave.
It seems to that very many people have all manner of issues from their pasts that partners accommodate. There are people with agoraphobia, social anxiety, OCD, manic depression and so on who are accommodated in quite extreme demands because their partner accepts their past. There are also people who have quite extreme ways of living due to religion. I don't consider those demands abusive, but I also think it is ethical for a person to leave such a situation if they can no longer deal with the way the other person has to live their life.
What I do think is unethical is to plan a pregnancy with a woman having agreed to certain ways of living she needs as a coping mechanism, and then announce during the pregnancy you don't intend to abide by them, but still intend to remain within the relationship.
What is the woman supposed to do? She is already pregnant. She cannot go back on the decision to be pregnant; she is in a vulnerable situation. The advice to get counselling may work, but maybe the OP has already had counselling. I know plenty of people who just live with their issues day by day, and have done for decades, because frequently counselling does not work.
I think you need to know why he has changed his mind at this point. Why didn't he mention this before you got pregnant. Because even if people do think your request for him not to be included it is not okay to deal with an abusive situation by abusing back, especially not if you live with kids as well.
Abuse isn't about requesting something, howver unreasonable we may consider the request to be. He can get his name on that certificate with mo further reference to you. You can't control that unless you are also doing something abusive to enforce it. Abuse is about having the ability to enforce requests and using that ability. You can deprive someone of money if you are the main breadwinner, you can prevent someone from leaving through financial control, you can socially isolate someone by refusing to do any child are of have a babysitter in the house. You has no power to enforce your request, as far as we know.
So the key here is why now? Because if the why now is, I lied to you and manipulated you into believing I agreed with you, but really I was just waiting until you got pregnant so that I could do the opposite knowing it was too late for you to back out, then you really can't trust him. Because that would mean he is prepared to use pregnancy as a means of controlling you.
But if that isn't the reason, and there could be others - a sudden death or hospital stay of a close family member made him re-evaluate family arrangements for example - you can probably talk things through and work that out so you both felt loved and respected.
But agreeing to a major thing with a woman, and then announcing you have no intention of sticking to it once she's pregnant and it is too late for her to back out, always concerning, whatever the thing agreed to was. And of course, one of the most popular ways of abusing women ever. So I'd want to know if I were you, is this the start of a pattern of controlling you through pregnancy and children, or is it just very unfortunate timing.