OK lets see if we can get the relay to work.
Connect the 2 wires you have taped to get the unit to fire to the output of the relay ( this would be the contacts/switch side).
Now the coil/control side of the relay connect one pole/contact to ground and take a wire with 12VDC and touch the other contact on the coil, does it fire?
Now that the relay works, we need to find a way to control the relay. I do not think the auto relay can be driven by the computer directly, not enough current, they take like an 1 amp.
I am not familiar with G540 is there an output that will drive the relay? If yes I would put a volt meter on it and see if you can control it with the computer. Once you find the right control to make the output happen when you want it then tie that output to the relay where you were touching with the 12VDC wire. The 12VDC wire was for test only.. DO NOT TOUCH TO THE OUTPUT OF THE G540 . The other side of the relay is still tied to ground. Hope this helps.