I'm not going to send you my personal info, but it's no secret that anyone with US health first gets a bill with a huge number $$ that's marked THIS IS NOT A BILL, then some weeks later we get another one from the insurance company which shows that number, and then a much smaller number "allowed" by the insurance which then shows as paid by them, then whatever "copay" we might have, usually about $30 with my insurance. Then there's "your responsibility" which is just the copay if the doctor/hospital is in your insurance network. In that situation the provider has to accept what the insurance allows and no more. If they're out of network, you may have to pay the difference between the original bill and what the insurance paid. Which can be a lot, so the first thing you should do is request an itemized bill. This will often be lower to start with, and gives you information to negotiate/contest items. Hospitals will also work with you to make a payment plan at that point.

In the OP case it was Swiss travel insurance not US health insurance and they finally agreed to pay the NICU bill, good work by the couple's lawyer.