Thanks for your question. We have passed it around our networks and hope you get some useful responses.
If you can't find a way to have the operation in a public hospital in Melbourne, one option could be to travel to Sydney, if a suitable surgeon who works in the public system can be found.
You could contact The Gender Centre in Sydney to see if they know of any surgeons. FTM Australia might also be able to help. Both of these organisations are listed in ZBGC's Resources Directory.
We wish you the very best of luck.