Surprise $4,200 bill after "in-network" surgery - anyone dealt with this?

Had knee surgery in October at an in-network hospital with an in-network surgeon. Just got a bill for $4,200 from the anesthesiologist who was apparently out-of-network. Nobody told me this was even a possibility. I asked the hospital beforehand if everything would be covered and they said yes.

I filed a dispute with my insurance and they said the No Surprises Act applies but then came back saying there's some exception I don't really understand. The anesthesiologist's billing company won't budge at all - they're saying I owe the full amount.

Has anyone successfully fought one of these surprise bills? I keep getting bounced between my insurance and the billing company. Feel like im going in circles. Did the No Surprises Act actually help anyone or is it just a feel-good law with a million loopholes?