I was initially inspired by [dmitry]s blog to see if I could maybe connect to the unit via it's UART and enable pairing-mode from there.

I was able to find a relevant datasheet for the BT SoC (BK3256) and connected to the UART test-points (marked URX & UTX on the silkscreen).

While doing this, I noticed that TP4 had a tiny label "ONKEY".. Could this be the button pad?

Pulling this high with the device powered got me a flashing LED and the device popped up in my BT devices list! Success!

I added a tiny switch from the parts bin, and with some hot-glue and a minor case-mod effected a working paring button.

It is not uncommon for this 'throwaway' hardware to be shipped unfinished or in an unusable state, so I hope this simple fix helps save something else from the landfill/back of the drawer.