In my state if the father is not present you can not put his name on it specially if you all are not married. I was not married to my sons father and when I was given the BC he was not there so I could not write his or sign his name. You probably do not have to put his name on it but it doesn't change the fact that he's the father. So you might have to ask someone in vital records in your hometown how that works.
They can just do a paternity test for child support and doesn't have to have name on BC. I kno from my situation.
If he is immature like that don't put him. I didn't put my oldest sons father on his cus he didn't want n wasn't around but I added him once he came back when my son was 2 and has been very involved since. But def don't put ur bd on the BC if u think he will try do something.
if you are not married then you have that option not to list the fathers name on the birth certificate, double check your state laws to be sure but most states if not all give you that option
If he wants to be on it and you are going to let her see him and you want child support then yes, if you aren't asking for child support or him in her life at all then no, just don't mention when you go into labor so he doesn't show up to sign it and stop talking to him unless he chooses to take you to court (which he probably won't)