Technically that is an option, however her OSHC will only cover the cost-equivalent of a public stay. If we make a private booking we have to foot the bill for the difference in the cost, which could run into many thousands of dollars. As a result the hospitals want us to pay that difference upfront (we've been quoted upwards of $12,000). We've recently bought a car, and are about to drop $4,575 on a visa application - we can't afford many thousands more upfront
Our GP, who has Obstetrician qualifications and has been looking after our antenatal check ups in lieu of having a hospital based OB, is livid. He says that just two years ago this wouldn't have even been an issue - that we'd have a booking no trouble. After all, we have the mandatory cover; it covers public bookings; therefore we should be able to get a public booking!
Edit: Our GP also said that as a last resort he can get us in as an 'emergency' if this drags out until she goes into labor. Her OSHC covers all emergency treatment, so at the end of the day we will get her into a hospital, but we'd really really like to get her in the normal, calm way and get access to antenatal classes and so on.