I'm not a health-care professional, so I don't know: But wouldn't you be able to not tell the hospital that you have insurance? Pretend like you don't have any from the beginning? Or can this backfire?
The practices we had our issues w/ were an imaging clinic and a pediatrician. I kinda expect that telling either one of them that we didn't have insurance would have resulted in them refusing to take us as patients. My wife and I talked about trying it with the prospective next provider, actually.