Will your baby be an Australian citizen anyway? I thought (and I could be wrong) that one parent needed to be an Aust citizen, not just a permanent resident. Apologies if I'm wrong, it's been a few years since we had to deal with any of this.
And without medicare you'll be paying a large sum of money regardless of if you are public or private. I can't remember how much it cost my SIL to stay one night, but it was super expensive. Lucky for them all was low risk so no NICU charges etc. But it cost them thousands.