Go public! The costs of going semi-private (or private) without insurance are impossible to calculate beforehand. You could have a straightforward birth and be out in a day and a half, or need a c-section and five nights of accommodation which will not be cheap as every single thing will add up. Also if baby needs intensive medical care (hopefully not!) then you'd be classed as a private patient for that too and those costs could spiral, but if you're public then baby is too.
That said, for some reason 4K sprung to mind when I first read the question (don't hold me to that!) and you can claim 20% tax back.