This is one situation where I would be hesitant to pay upfront, as I do believe that billing departments work harded if they haven't been paid. I have had situations where my insurance company has been slow to pay and I have been hassled and could have paid with a credit card (even threated with being sent to collections), but I held firm saying that this should be paid by insurance. I'm afraid that non-payment, sometimes is the only leverage you have. I would have to be absolutely sure. Also, many hospitals/dr.s are suppose to "write-off" amounts above the "allowed amount." If you have paid in advance, they might not and you would have no recourse. It's great that it has worked for you.
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DtG