The module is intended to have volume adjusted on your playback device, so does not have a volume control directly on it.(the buttons just send a signal back to your phone to adjust it's volume)  The bluetooth module puts out a beep every time you adjust the volume.  I intend to use this a room sound which will be faded in and out, so those beeps are not workable.  Combined with the bluetooth connecting sound and the speaker pops when powered on, I decided that the simplest solution would be to insert a volume control into the audio path.  This would allow me to just turn it all the way down when connecting, and then do my fading with the knob instead of my phone.

I knew from the product picture and the information I could find about the components, that audio comes from the BTM835 module, through capacitors and into the TPA3116 chips.  I figured I could just insert a stereo potentiometer between those.  One concern I had was that the audio was differential.  To proceed, I needed to trace out a bit of the schematic from the board.

The output of the BTM835 module is indeed differential, and all 4 of those lines go through capacitors.  However, the other side of those capacitors do not go directly to the TPA3116.  Instead, they are fed into a dual op-amp, which is providing some gain, but also converting the single to single ended.  This circuit is powered from a 12V LDO(which means it'll run lower if the main input is 12V), with the audio biased to 6V so a single ended supply could be used.  I am not sure why they convert the audio to single ended, given that the TPA3116 has a differential input(which can be used single ended.  Perhaps the output of the bluethooth module was not enough, though with a configured op-amp gain of only 1.3x, I'm surprised.

The output of the op-amps go to the not populated connector marked L - GND - R, and then under the heatsink.  I can see the traces connect to some ceramic capacitors, but I cannot confirm if they connect elsewhere.

Measuring the output of the op-amps with a scope, I saw that the audio gets to about +/- 750mV, with a 6V common mode.  This 1.5Vp-p is pretty much line level, though the 6V common mode offset means it wouldn't be a good idea to directly use this signal as a line out from the bluetooth receiver.(though could be fed through capacitors to go out)

The end result of this is that I can easily install a potentiometer as a volume knob, after the op-amp.  I separated the op-amp output from the amplifier input by cutting the bottom side traces.  Given the 6V common mode, I wired one end of the pot to the 6V rail, the other to the op-amp output and the wiper to the amplifier input.