I finally opened one of my universal modules tonight to investigate. This one is a PUM01 with date code 10F23. Before opening it, I connected it in place of the UM506 in my first post and confirmed that it behaved the same way.
The initial inspection reveals that every pin of the MCU is connected.
The MCU is a Sonix SN8P2501BSB. I found a datasheet, which I attached here, and a photo I took of the PCB.
I suspected it was possible that pin 13, which I thought was leading to the pad in the center of the OFF dome switch on the top right, was picking up a spike since it did not appear to be pulled up or down. At the risk of disabling the button, I tried pulling pin 13 both up to Vdd pin 4, then down to Vss pin 11 with a 10KΩ resistor with no success. It was then that I realized that the trace for pin 13 was actually leading to a transistor between the ON and OFF pushbutton switches at the top, which was in turn connected to the pads on both of them.
It also would not make sense that the MCU is picking up an errant OFF signal, because as I mentioned in my first post, the sounder does not stop beeping when the anomaly occurs.
At any rate, without a schematic or a clear understanding of how this circuit works, I'm not sure what to try next.
I hope Authinx plans to redesign these!