@ogden - thank you for your help Were you able to see the photos in the download link? I'll try and down-res them so they can be uploaded directly.
Yes, it worked for me. Just took extra effort. It's good that you uploaded images here - for those who are just reading this thread.
The middle contact on all the switches are connected to each other and a wire (black) is run into one of the terminals in J16.
Black sounds to me like ground. You could measure voltage on all three terminals of single switch which is not pressed. It will be enough.
So each existing switch would have a diode soldered to each solder lug and (pending testing)
Yes. If black wire is ground then you solder anode end of the diode to lug with color wire, each switch receives two diodes. I would suggest small generic 0.1A Schottky diodes like BAT42. Big high current rectifier diodes could have big forward voltage which may render microcontroller unhappy with resulting level of the signal received.
the new switch would daisy-chain
Not daisy-chain. New switch receives cathode ends of diodes, four wires from existing switches upper diodes to upper lug and four wires accordingly to lower lug. So when you press new switch down, it pulls all lower wires of existing four switches to ground through diodes which are required because that pulling shall go only one way (diode does exactly that, "one way").
[edit] I have no mood nor time to draw even such a simple schematics. If you need one - try to make it happen and show. Then I can confirm - it's ok or not.
[edit] Forgot to
say repeat that middle lug of new switch shall be connected to mid-lugs of other switches.