EDIT: This Sound Town unit might work better for you. Coming from Amazon Australia, so probably has the necessary power plug. And from what I can understand the microphones for this connects via Bluetooth, so shouldn't have a problem with wireless frequencies making your neighbours mad.

From the information on this I suspect that if you use the Audio In port then that might override the HDMI audio, so you probably can't use it the way I suggested earlier. That is to say mixing audio from HDMI with Audio In.

With this you would sell the JBL wireless microphones and use these instead. Not sure if they are any good though..