Insane billing after child birth

My wife gave birth mid 2021. We purchased the best health insurance available to us at $850/month in cash. On paper, an absolutely stellar plan.

We researched everything meticulously, ensured everything possible would be in network.

Over the past few days we have received $10,000 in balance billing due to rejected claims by the insurance provider. This is absolutely insane. It was an uncomplicated child birth in every single possible regard, and all of the services we received were discussed in advance with the hospital and insurance provider.

We are completely defeated. A cash price for an uncomplicated child birth at this hospital was $13,000, which is significantly less than the insurance and all the billing we have received/payed. In fact, WAY less.

What can we do to fight this that is not going to consume hundreds of hours of our life? We now know medical insurance is a scam to a very high degree. Even if we could afford a payment plan to pay this off, we have reservations paying into such a corrupt system; perhaps we should let it hit our credit and wait 7 years? Any guidance navigating this is helpful.