In-network hospital, in-network surgeon, but the anesthesiologist was out-of-network - now I owe $3,100

I scheduled a knee arthroscopy in January. Called my insurance ahead of time to verify the hospital and my surgeon were both in-network. They were. I thought I'd done everything right. The procedure went fine and I came home expecting my normal cost-sharing to apply.

Six weeks later I got a bill from an anesthesiology group I'd never heard of for $4,800. My insurance processed it as out-of-network and applied my out-of-network deductible and coinsurance, leaving me with $3,100 owed. Turns out the anesthesiologist who was assigned to my case was not in my network, even though literally everyone else involved was. Nobody told me. Nobody gave me a choice. I couldn't have made a different decision even if I'd wanted to.

I've read about the No Surprises Act and I think this should be covered under it - this is exactly the scenario it was designed for, right? Scheduled procedure, in-network facility, provider assigned without my knowledge. My insurance is pushing back and saying the NSA doesn't apply because anesthesiologists are "ancillary providers" and there's some exception. That doesn't sound right to me.

Has anyone actually won a dispute like this? What's the process? Do I need to involve my state insurance commissioner or can I handle it through the federal No Surprises Act dispute process?