I deleted the unknown device then let Dell's diagnostic software look at the machine for missing software and updated the driver for this WiFi/Bluetooth card. Now the Windows Update has happened again and this time the Bluetooth appears to be okay.

It took a few days to realize that something was wrong as far as the computer, not just a case of defunct headphones, plus 24 hours of fussing with the computer to fix it. No dust disturbed in the process, several restarts and aside from a couple of hours lost in that one system restore, it's a good outcome.