The collar uses a 40 kHz ultrasonic transmitter. I attempted placing a transmitter emitting a 40 kHz pure sine wave tone near the door, but that just resulted in an LED blinking to indicate interference.  It turns out that the system has some protection against false positives built in by pulsing the 40 kHz carrier. Hooking up an oscilloscope to the terminals of the collar transmitter, I was able to observe that it sends out 40 kHz pulse trains of 0.888 ms long, at a 5.84 V amplitude, and pulse trains occurring at a frequency of  38.8 Hz. 

This sounds like a job for a 556 (the 16-pin dual 555 variant). My current plan is to use one of these for pulse train generation and avoid the need to deal with interrupts. High-level control will be provided with a NodeMCU that interfaces with a PIR sensor.