There will be a gap between the fee for your anaesthetic and the rebates from Medicare and your insurer. It would be more accurate to call it an insurance shortfall and it is the direct result of federal government policy.
Each anaesthetist sets their own fees. As a general guideline, for most procedures, your anaesthetic fee will be in excess of the Medicare rebate but less than the fee that the Australian Medical Association recommend as being fair and reasonable. You will usually receive a written estimate of your anaesthetic fee. Uninsured patients are normally requested to pay their account in advance.
If you would like to learn more about the Health insurance rebates for anaesthesia procedures – this link from the Australian Society of Anaesthesia regarding billing.