... While an appliance module clicks on and off.
I wonder if the appliance module is actually clicking
OFF and
ON instead...
If the sensor is set to OPEN the circuit when the door is opened, and CLOSE the circuit when the door is closed (which is a little more common for magnetic door sensors), then the PowerFlash would be sending an OFF when the door is opened.
The "click" of the appliance module sounds the same for both "On" and "Off," so unless you plug something in, you might not realize if it is working backward.
If that's the case (the sensor is a "normally closed" one, and is opening the circuit when the door opens), then the Powerflash won't be able to directly control the chime. You would either need to swap the sensor for a "normally open" one, or write a macro in AHP to sound the chime based on the "off" command.
What HouseCode / UnitCode combination do you have everything set for?
Do you have any other plug-in controllers (or a TM751 and a hand-held remote) that you can test with?
Maybe the PowerFlash isn't working correctly.