Got a chance to do some quick tests. Seems that it sends a small current through the bulb to keep the time delay circuit from retripping.

Figuring a 120ohm (roughly 1.2 watt bulb resistance, while lit), 5+ watt resistor mounted on a heatsink should do it, though the led bulb I selected seems to be particularly good with lighting at very low voltages and may require a zener diode or some such.

The easy solution would be to disable the light or leave in an incandescent bulb, but I am fairly dead set on installing LEDs.