It is against the law to change or cause a child to be known by a different name without the agreement of all those with PR. We have established your ex has PR because he is on the birth certificate and your daughter was born after the law was changed.
You can not just call your daughter something else, or tell schools, doctors etc that her name is your surname or any other name, even double barrelled. Well you can but your ex just has to show them the birth certificate and they will change it to what that says without any further discussion. The space on doctors forms etc is for people who have changed their name legitimately, either through marriage, deed poll etc.
Getting your child's name changed is a battle you won't win. This is one area where the courts rule in favour of the father. Pick any battles wisely. This isn't one to attempt.
I know lots of children who have different surnames to their mum, where they're not married but have their dad's surname regardless. It's not unusual. You could always change your surname to be the same as hers if you are that worried about the name difference. But you wouldn't want to do that i'm sure, just like dad isn't going to allow you to change his child's name. It's the one thing she has of his that you can't attempt to control or take away from him, or from her for that matter.
You might not like your ex now, and you might want rid of all trace of him, but without him you wouldn't have this beautiful wonderful child. It doesn't matter what her name is, she is still the same child. Think about what's important here. Maintaining a responable relationship for the sake of your child is important.
Oh and 9 months old is more than old enough for contact with her father. At that age your daughter should be seeing her dad frequently for short times to maintain and build a good relationship. Your daughter has a right to have a relationship with both her parents. Don't put blocks in the way just because you find it hard. Do you leave her with your mum, sister other babysitter ever? Then she is old enough to spend time with her dad too.
Try not to think too much about how much you loathe him, concentrate on how much you love your daughter and do what is best for her.
Gilly