One of the very first things I taught my daughters when they got their license is to call 911 if they don't feel safe, and I reiterated that same lesson after the mass-shooting spree here not long ago. I feel that same lesson would work here.
I taught all my kids, but mainly my daughters, that if you're getting pulled over, marked vehicle or not, if you don't feel safe and you don't feel like it's a real cop, just call 911. The dispatcher can reach the driver, and verify it's a real cop. Don't speed up, just drive as normal until they get the okay from the operator that it is a real cop then pull over. Yes, in the end, you may up with an extra charge for failing to stop, but that's for the courts and a judge to decide if it comes to that. A phone call by the "Doctor" to 911 could have avoided this incident entirely, or at least diffused the situation to get the lady to the hospital asap.
I assume most police officers would escort them to the hospital immediately, and sort out the validity or bullshit afterwards. This isn't a patient, it's the doctor. I believe they would hate to be the reason for the delay that could cost a life.
Fun fact... most cops aren't dicks if you don't treat them like dicks. Some are, but most aren't.