I power the Bluetooth receiver/adapter from a USB wall charger.
Am currently playing around with a DROC CA200367 Bluetooth car radio adapter sold on Amazon here:
It has two advantages: first, it can run on 5 – 35 volts, allowing a wide range of ‘wall warts’ to power it, and second, it connects automatically with a Bluetooth transmitter, without having to push any buttons. Unfortunately, it draws about 45 mA and so I’ve not been able to find a place within an AA5 to ‘steal’ this much DC low-voltage power.
Rick Williams in Edmonton has pointed me to an even more sophisticated FM/Bluetooth adapter for car radios, described here: