[...]
> Anyway, the device "un-pairs" periodically. And, when this
> is discovered by the residents (me!), refuses to pair until
> it is power cycled.
> [...]
> Let's assume you can't AUTOMATICALLY re-pair when you
> rediscover the phone (gee, my car manages to do that!).
After it's power-cycled. More than once have been on a road trip where my phone's BT connection to the car (for music or nav) has dropped from the car-side, an and the only fix is "get off at the nearest rest-stop".
Granted, my car's a decade old (or a bit older); so maybe a newer car would behave better.