D.B.
I think you have 2 things to do right now. 1) Have a safe delivery of healthy babies. 2) Take care of yourself emotionally and physically.
You have had many months away from a controlling man who emotionally abused you. Good for you! You have managed on your own and that is a tribute to your strength and good sense. He has threatened you. I would not do a thing with him right now.
I would block him and seriously check your privacy settings on Facebook if you are still using it. I would be very careful about who you bring into your life as friends and make sure they are loyal to you and you alone. Surround yourself with good and supportive friends and family members whose only concern now is helping you out with new babies.
Whatever you need to do to enjoy your children, you should do. When you get through the first few months of sleepless nights and hormonal changes, you can take your time finding a family law attorney to find out what your options are. You do not need to put your ex on the birth certificate right now. You can also call the National Domestic Violence hotline - even though your ex did not physically hurt you. They are wonderful counselors with all kinds of information and resources available to you, free of charge. They are experts in all of these areas including custody and child support and safe visitation (or the withholding of visitation). They will be able to help you right now about what you need to do in the hospital and how you can protect yourself from him if he somehow finds out. You have legal rights. Find out what they are.
Take some time to decide what you mean by "know who their father is." Do you mean they need to spend time with him, or that they need to know his name? (And you don't need to decide that right now.) There is no reason you need to have an abuser in your life if you feel you or your children will be threatened.
I would no make his decision while hormonal and stressed out and tired. And I wouldn't do it alone - have knowledgable experts in your corner before you do anything at all.
Good luck - wishing you a smooth delivery and much joy with your babies!