I would have taken the cops badge number and broadcast it to the 911 operator if the cop did not want to be recorded and I would have asked the other paramedic to identify himself to the 911 operator with his license number (I am assuming the state has licensing provisions for accredited paramedics.) and then stood by to give facts to the first responders on the scene.

Not being and EMT or paramedic, I would not have been able to help to the extent you did, but I have a strong aversion to people who claim false or exaggerated credentials.

Bountyhunter