I think I can mark this as solved. The answer is the best one for me, but it doesn’t answer the original question… read lots of the other awesome replies to get at that if needed.
As always, the simplest way tends to be the best. Since the door sensor is just a pressure switch (open door = open circuit), I decided to stop messing around with the stupid alarm panel since I’m not really using it anyways.
So I just grabbed the two wires (to_door and from_door) and plugged them into ports on the Core such that to_door is digitalWrite’ing HIGH and from_door is set to digitalRead. Door closed = HIGH, Door open = LOW. (I probably should be using a pull resistor, but it seems to be working fine without one.)
Some day down the road I may figure out a way to make it work with the built in alarm panels, but right now that’s pretty low on the list of priorities.
This has been my first go-round with the Particle community, and I want to thank everyone for assisting me in this project 
you all rock. I look forward to doing more fun stuff to come. Once I get it all connected to homeassistant, I’ll post a write up somewhere.