I don't have a NAPA number.
The relay can certainly click and still be bad (e.g. burned contacts). I guess I was putting a lot of faith in the relay you substituted and claims by others that this is a common automotive relay. There is another possibility and that's a bad wire or connection, either feeding the relay or between the relay and the starter.
Do you have a voltmeter/multimeter and and experience usinga schematic? I don't have the exact schematic for your model but I have similar ones. If I were doing this and had a meter, I'd first test the voltage at the small starter terminal, which should jump to 12 volts when you turn the key to start with the clutch pressed. If that tests good we have been looking in the wrong place so far. Assuming that tests bad (like near zero volts at the starter) I'd test the contact connections (not the coil connections) at the unplugged relay connector. One should read about 12 volts and the other should be zero. Jumpering across these two should crank the engine regardless of the key switch or clutch switch. Don't try this unless you are sure you know which are the connections for the contacts . If not, it's either the wrong connector or there is a wire problem. You can ensure it's the right connector by checking continuity between the small starter terminal and the start relay contact on the connector that was zero volts.
The push-button you proposed (from 12v to the small starter terminal) will work for a while. The problem is the sparking (which you no doubt observed) and fairly high current that will eventually burn out a push-button switch unless it's designed to handle that current and arcing.
Fixing wireing problems is very messy so it's best to test everything else first and do enough testing that you know exactly which wire is bad. Often it's better to just run a new bypass wire than to tear into a factory cable bundle unless you have multiple failures (sometimes caused by rodents.
