I did some more testing and I learned this "rapid clicking" is the work of the thermostat. I assume some sort of cut off due to not getting the responses it wants.
I learned this by (finally) learning I can bypass the thermostat outright by jumping Y (compressor) Green (high fan) and Red (12v).
When I do this I do not get clicking and the unit stays on but I do not get cold air. I only just started messing with it in this configuration but I noticed it did sound like the compressor would periodically try running (hums) and then gives up when I bypass the thermostat.
A cursory search leads me to believe this is the capacitors but I replaced the "Start" and "Run" capacitors with new ones that can hold quite a bit more. I didn't swap the "Fan" capacitor but the fan works fine.
EDIT: Swapped fan capacitor as well, seemingly no change.
