The electronics inside the BPCs would probably prefer to have a direct feed from a transformer without going through something that chops up the waveform like the ZW and CW-80 do. Do you have a suitable wallwart?
Since one BPC works and the other buzzes, my first suspicion is that the energy storage capacitor in the power supply of the buzzing unit has gone bad or is disconnected. If you were to open the unit, the capacitor would be obvious since it is the largest capacitor on the circuit board.
It is the big can on the right edge of the board. I can't read the identification in this photo, but it is probably "C3". The other possibility is a bad diode.
How old is your unit? These capacitors can fail due to age.