I've seen this happen in California as an insurance agent. For instance, one of our clients had to have emergency surgery and went to a contracted hospital. She could have fainted when she got the bill from the anesthesiologist that was on call that night whose office said they would not accept her insurance. She had no control over which anesthesiologist she got. I got it billed as an out of network referral and the insurance paid the contracted rate for what were considered usual, customary, and reasonable charges. The anesthesiologist accepted it as payment in full.
I think the trick is to have a good broker to deal with the insurance carrier or get someone good in claims to assist you with these things. Sometimes you just have to know what to say.
In the case of an ER visit, you have no choice of who is on duty. Ask that it be billed as an out of network referral and usually the provider will accept the payment from your insurance.
I was successful doing this many times with Blue Cross/Blue Shield.
Document every conversation and correspondence including the date, time and person you talked to. Get their extension. If you can't get anywhere with that person, talk to someone else.
Like I said, if you have an insurance broker or agent who can do this on your behalf, that's all the better.