it's a disgrace. Go public or enter into very brisk negotiations with the service provider. They are generally very driven by finances and will accept a little less rather than miss out on accessing your wallet altogether. You need to be as ruthless as they are.
Up here it's the same mob anyway, I can't understand why anyone would go private to get exactly the same treatment in the same facility. You can choose to go elsewhere, just check out your options and figure out what you are willing to forego in the way of convenience if you want to cut your costs. You have a right to access treatment in the nearest public facility even if you have private health.