My cousin tried to not put the father on her child's birth certificate and found out she wasn't entitled to a lot of centrelink benefits that she needed to live. I think this might have been because she was only 16 at the time and not living at home.
If the father wants to be involved with the child than she has no choice. He can go ahead with court orders to demand a DNA test. She would also not be entitled to any child support so not sure if it would actually be the best option for her. She will need to see a lawyer to find out what she can do to prevent contact if that's what she truly wants.