I had an emergency c-sect and had two anethetists (one did my epi on the labour ward and then I was transferred to theatre and had a different one) - both billed my PHI direct and I paid no OOP for that.
My total OOP for the pead was about $150 after medicare (not covered by PHI because DD was not admitted to hospital). Unfortunately, unless you have a preplanned elective you won't know who you get until the day making any expenses pot luck so to speak. However, you *should* have reached your medicare threshold so should get 80% of any fees back from them.
Don't forget that if you have a c-sect there could also be a bill for the surgeon's assistant (you will have two surgeons - your Ob and another Ob that assists them). Mine was about $200 OOP for that.
You could contact the hospital and find out which anethetists work from there and then contact their rooms and ask for estimates based on the scenario you think you are likely to be in. Or ask your Ob who they tend to use for c-sect - some have anethetists and peads they use (even in "emergency" cases) and might be able to give you an idea who is likely to be "on-call" around the time you're due. The anethetist in the delivery suite though is usually whoever the mids can get hold of.

If it helps, my TOTAL OOP (surgeon fees, paed, pathology etc) for DD's delivery was less than $500 and this included paying for some path stuff for her as well.

HTH

MG