You can't by law. Of course you could pay the whole sum yourself and give fake details.

You by law have to provide your details and it is listed on your Medicare. I found out the hard way

You can’t get one without giving your details. You can make sure your Medicare is set up so it doesn’t get sent to “My Health Record”, and if you are worried about it showing on bank statements you might need to open a second account. When you book you can discuss all of this with the operator, they are really helpful.

You can’t keep it off the books but your privacy is maintained by law providing your not a minor

Try Marie Stopes