UPDATE & presumed resolution
Medicare just processed the private ambulance bill. The submitted bill was for $939.29, of which $860.99 was for a "private ambulance", and $78.30 was for mileage. (The distance driven was 3 miles.) Medicare approved $273.22 for the ambulance, and $22.86 for mileage. Medicare paid 80% of these approved amounts, and I expect my mom's Medigap supplemental to pay the remaining 20%.
I noticed that Medicare approved a higher amount for the local government ambulance which took her to the hospital because that was deemed "emergency" transport, whereas her private ambulance trip home was after successful surgery and was not considered emergency transport. Indeed, it was much easier to move mom in and out of the ambulance after the surgery.
I did not sign the ambulance company's misleading "Signature Request Form", and never contacted them regarding their form/bill. But as I previously wrote, I contacted the hospital and they sent (or re-sent, I'll never know) a "Certificate of Medical Necessity" to the private ambulance company.
I may send a copy of the "Signature Request Form" to my mom's state Attorney General's office to complain about the sleazy manner in which the ambulance company attempted to get a signature under false pretenses from a 99 year old woman, in which she would have agreed to be responsible for the entire highly inflated bill. The Signature Request Form falsely states in all caps: "SIGNATURE REQUIRED IN ORDER TO FILE AMBULANCE CLAIM".