Welcome @mellymell. These can be hard choices. You say you have health insurance and that you're new to Australia. Does your country of origin have reciprocal arrangements so that you can access our public system? If not, you may still face put of pocket costs even if going to a public hospital. Most large public hospitals will have someone you could talk to about these questions
I did a mix of public and private, having chemo in the public system, followed by surgery in the private system, then radiotherapy as a public patient. I can't speak for Melbourne, but where I am, the private hospitals don't have the support services that the public hospitals do. However, the private hospital i Sydney where I had my surgery has amazing supports for inpatient care, and wonderful breast care nurses.
A coyple of things to think about in terms of any future surgery. Would you want the same surgeon? If so, does he/she work in the public sector anywhere. If you wait for furthet surgery until after the rest of your treatment, whatever that entails, it may be classified as elective and ad such you could potentially face an extended wait, especially if reconstruction is a consideration. If you opted for private radiotherapy if required, there could be considerable out of pocket costs but there are also some centres that will negotiate either the cost, or a payment plan.
Seven years down the track, I have still got ongoing costs as a result of my bc but that's a whole other story.