FYI
On both of my childrens birth (married) my husband was in the room each time when the nurse came in to have me fill out the birth certificate. Only I was allowed to do this for listing name for baby and name of father. They don't even look at dad. I asked the nurse about this later after the second time. Only mom is allowed to fill it out. I could have put under father, unknown or Santa Clause. I don't know what happens if the father brings it to court. I suppose after a DNA test for paternity he could have his name on the Birth Certificate, I have heard of this being done. Does your daughter really want to go through all that. Or do you think dad will skip out later.
My sister actually did that, put father unknown. She wasn't married, father was there for birth but guy was really a loser and did take off 3 months after birth. There was actually a fight at my parents and the guy had baby in his arms and was going to take off with baby. When the police were called they asked if he was listed as the father on the birth certificate, no he wasn't, The police made him hand the baby to my sister or he would be charged with kidnapping. If he was listed as father the police could not have done anything since it's not kidnapping if it is your child. It was also easier for my sisters husband to adopt her son when she got married to different guy 3 years later.