How do you negotiate charges with medical providers?

One of my young adult children recently made three office visits to a medical provider, none of which exceeded an hour and none of which involved any "hands-on" medical procedures; they may have taken his weight/temp/BP/pulse but the remainder of the visits were basically consultation/counseling. The son in question has health insurance through both his own employer and through mine. However, the medical provider in question seems to be using an approach of "set the prices so high that they will exceed any insurer's covered amount so that we can get the maximum payout". In this case, the provider is trying to charge about $3700 for three office visits. Negotiated discounts with one of the insurers will drop it by 10% and the combined payout of both insurers is about $1400, leaving about $1900 outstanding that the medical provider may be billing us for. This exceeds by a huge amount the copay we might reasonably have expected to owe for services of this nature (I would have thought $50 per visit would be reasonable if a little steep; anecdotally, my son claims he was told would be covered by insurance and would cost little or nothing out of pocket).

Some recent news stories on medical costs indicate that medical charges of this nature should not be treated as a final amount owed, but as a starting point for negotiation. Frankly, I strongly dislike the fact that I am in this position, but it may well be that the only way to avoid paying the full remaining balance is to negotiate. I'm worried that having already negotiated 10% off with one of the insurance carriers they may be unwilling or less willing to negotiate further discounts with me. I have no idea how to approach this. Can anyone offer any suggestions on how best to engage in negotiations of this nature with a medical provider?