I went private for radiation on Central Coast NSW.
The only reason was that I told my surgeon (private) that if it was going to cost more than $500, I would switch to public.
She organised for me to go to the radiation clinic in their "team" who agreed to do the 25 cycles for under $400.
I received an account to take to Medicare on two occasions, they gave me a cheque and I paid the difference. I understand from then on I was bulk billed.
Strangely enough, when I took the two accounts with the out of pockets I had paid ($394) to Medicare, Medicare then refunded an additional $220. I have no idea why, but I was grateful for it.
Apparently you can negotiate with the private clinics, but it helps to have your surgeon on side, and for them to be working in a team arrangement.
I guess it also helped that there was a big Cancer Centre at the public hospital on the other side of the railway line, so they had competition.