Thanks for the tips.
In the end my wife visited the DIBP office in Sydney, who told us what the process is. It's actually very simple and easily done yourself, assuming the child was born in Australia, and you haven't yet left the country with him or her.
For anyone else who needs to know:
- Get Form 1022 and fill it in
- Get scans of:
- The child's birth cert
- The child's passport if you have it.
- The parents passports
- A copy of the visa grant notification for your 457
- Email everything to 457@border.gov.au. Include an explanation of what you need and your circumstances. If it's urgent, then tell them.
Our request was processed in about 3 working days.
When you email the request, you receive a response with lots of helpful information on how to do various things related to your 457 visa, including this information. It's a shame the information isn't available on the web, so I'm posting this in the hope that it will help someone else in our situation. One agent quoted us $800 to do this for us, which is frankly a huge amount of money to fill in a simple form and send an email. About 15 minutes work in total!