Yes, this is exactly what I'm doing. The HC12 adds complexity, but it's really just a wireless LED on D12 that turns on when there is motion. When either the 12V GND OR the +12V touches the circuit, the LED indicating that there is motion turns on showing that the PIR output is high. The pulldown resistor stabilizes the PIR output and the LED is turning off/on as expected (with the 12V power connected). When I introduce delay so that the PIR isn't flipping the LED real time, I see that this IS actually working perfectly.
I do wonder why the house-connected 12v supply would affect things this way.