Wiring a bluetooth module into car radio, issue with switching between radio and bluetooth
I am assuming that the two sources playing at the same time are the radio and the BT stream.

Looking at the schematics at the same page 5, but in the bottom middle of the page, there is a section called LW Mini 14, this is the mechanical tape control part. On pin 1 of the connector, you can clearly see a switch with the symbols of tape and radio. this switch is the mechanical trigger to tell the system to switch from radio to tape.

If you take this pin to ground you should only hear one source, which should be BT or tape, if you have one playing.

If instead you have BT and the tape playing at the same time, then, all you need to do is to put a tape in the radio that doesn't have tape. so no sound will come out.