"On appeal, his insurance agreed to cover $9,967 of the charge—better than nothing, but it still left him on the hook for about $3,000."
Tell the insurance company to wait to release the funds until you talk to the ambulance company or to release the funds directly to you.
Tell the ambulance company they can take the $9,967 or get zero.
This isn't sustainable. You really want every person who is dealing with this bureaucracy to manage negotiating between their insurance and their EMS? After they've just had a significant accident/health issue?
That seems needlessly cruel to do to people.