Bulk billing and medicare rebates are two different things.
It is up to the individual doctor as to whether they will bulk bill their patients for any particular procedure. You will find that some GPs will bulk bill for certain procedures where the medicare rebate is somewhat related to the actual cost of the procedure, such as routine consultations or some very minor surgical procedures. The problem with bulk billing arises because most medicare rebates are not an accurate reflection of the cost of the procedure; they are designed to offset the costs of a procedure, rather than pay for it completely. Bulk billing in these cases leaves the doctor our of pocket. I think that it is very unlikely that any doctor would bulk bill a circumcision, becauee as explained, the total cost of the procedure can go into the $1000.00s depending on the method of anaesthesia, whereas the rebate is only $1-200.
I'm not sure about private health insurance.

