I'm stepping a little into giving "advice" here, so I'll put in my usual disclaimer...

THIS IS NOT A SUBSTITUTE FOR INDEPENDENT LEGAL ADVICE. If you wish to obtain advice in relation to your particular situation, please contact your own lawyer.

I would not advise you to put your name on the birth papers. In Australia, this is an offence.

Contact your local Department of Human Services (child welfare) office and ask to speak to someone in the Adoption and Permanent Care Team. They can give you free advice as to the process required in your State.