Switching 1 BT Module between 4 Arduino devices
Another possible approach is using yet another micro as a router of sorts. Have all of the subsystems talk to it over UART, SPI, whathaveyou, and let this router handle the BT communications.

This way sure does have some benefits - you can keep the connection to the BT module at all times, which may even result in Better UX™; you can make it present a simpler/better-suited protocol to the other subsystems, without those needing to care about which particular BT module you happen to use, if any! Abstraction!

You seem to have no shortage of microcontrollers in the project already, so tossing another one in shouldn't be that much of a problem, eh? :D