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When I push in the clutch, I clearly hear a relay clicking somewhere every time. Does this mean the clutch safety switch is working?
That simply means something mechanical is moving and you can hear it. If you want to know if the physical switch is working. pull the connector, make up some jumpers with a battery powered light in it and plug it into the switch. If it goes on and off when you depress and release the clutch then the switch is likely good, if not then it's bad.
To bypass the switch, just make up a jumper that plugs in the connector that typically fits to the switch. That will act like the clutch is depressed so the car can start (all else working correctly). Remember though if you do this that the car can be started in gear now so be careful.