Here's a device that, when the bottom door key is covered/uncovered continuously, will porduce the opposite effect out the top. It's nowhere near as compact as I'd like, but it works. It's also about as fast as a door key chain up the length of the construct.
I have used a different device before, which works well if the door keys are horizontal, are the top one triggers the bottom one. Added to your device: fynatyg