Out-of-Network Doctor Showed Up During Delivery

I’ve been trying to fight a doctor bill from my wife’s delivery last year. I’ve established that since we have workplace health insurance and the delivery occurred in 2021 that I can’t use any Surprise Billing Law despite being in California. The federal law started in 2022 and the California law is only for state insurance plans.

The doctor (while in-network) was working through a provider (out-of-network) and assisted with the delivery. I received a ~$1000 bill for services (in-network deductible + OOP met for 2021 but OON deductible + OOP not met) and after a phone call the billing department told me to contact insurance and that “they might just pay it since you really have no say in who shows up during the delivery.” I doubt they will just pay it but is there anything I should mention when I call them to help my chances? Is a birth considered “urgent” or an emergency since you can’t really predict when it will happen? I had no way to verify the network status of the doctor in the delivery room. The delivery took place at an in-network hospital so there is no issue there and those bills have already been paid.