Before I build: will this circuit work, or have I overlooked something?
You should not be loading the analog signal lines with anything besides the input that the signal line is meant to drive, and in this case, a signal attenuator (the potentiometer). In other words, take those LEDs off those audio lines! 

Not that it will work anyway. For example, the audio line out from a PC or laptop or tat BLE adapter (the green 3.5mm jack) reaches at most 2Vpp.  So the signal won't even overcome the LED's voltage drop to turn them on.  But you should NOT have LEDs or anything else connected to a relatively high impedance analog signal line, and especially something non-linear like an LED.  And those audio signals you could plug into the jack or generated by the bluetooth adapter are intended to drive very high impedance (~10KΩ+) inputs, the typical input impedance for an amplifier.  For example, the amplifier you've picked has 22kΩ input impedance. 

If you want to indicate switch position, just get a DPDT switch and wire one pole to the 5V (or 12V) power line, and the LEDs with series resistors to ground on the two of that pole's throws.  Then of course wire your audio input to the other two throws and the output to the potentiometer to the second pole.  

Also, your potentiometer is connected incorrectly for volume control.  As it is wired, it is just a fixed resistor in series.  The resistance of between pins 1 and 3 never changes, if it's a 10kΩ pot, then you've wired it to just be a 10kΩ resistor in series with the audio signal.  Another tip with signals: you generally want the load being driven by them to be as fixed and unchanging as possible while you're actually playing audio.  Load changes will cause distortion.  In fact, that's how volume control works - by varying the load, but in a small and controlled fashion.

With that in mind, the correct way to wire the potentiometer is with pin 1 to the audio signal, pin 2 out to the amplifier, and pin 3 to ground.  This is seen as a 10kΩ resistor to ground by the audio signal. Depending on wiper position, the signal amplitude will vary.  



Also, I'd suggest powering the bluetooth dongle from a 5V linear regulator.  I doubt it uses more than 100mA, and it's possible the switching regulator could inject noise into the audio signal path.  

Finally, given the simplicity of this circuit, it is an ideal candidate for "star grounding".  Simply put, every component that has a terminal going to the - terminal of the battery ('ground') should have it's own wire just for that component that is connected at the battery - terminal.  All ground connections are made in this way, so all grounds connect at a single point (and it might even look kinda like a star).  This would remove the need for a ground loop isolator completely.