I was thinking of doing something similiar at one point but never got around to it. Thanks for posting about it. I'm really surprised they thought the BT needs to be on 24/7.If I remember well, when the BlueTooth module is disconnected we loose the voice commands for radio, A/C, etc.
What I did to keep the module connected, use voice commands, and avoid the battery drain, I installed a micro-relay that disconnects the power from the module when the car is OFF. The relay is powered by the AUX power and the Normally Open (NO) contact interrupts the permanent power wire. I used this wiring for several years without any problems. Even after I replaced the BT module years later, I still have this wiring configuration in place. If the new module goes bad ever, I will not have a battery drain.
The factory wiring keeps the BT module powered 24x7, that is part of the problem in the electronics failing. This relay wiring switches the module ON only when the car is used, therefore it should extend its life.