That relay can handle 240V. The coil to energize the relay takes max 24v. I have not completely read all of the diagrams you created, so I will look back through them. Using a relay is the EM approach to solving this design. I am not saying it won’t work, but it’s an old school approach.

I would also be concerned using the 12V switch matrix power to drive a relay. I don’t believe the circuit is designed to provide enough amps to trigger a coil in a relay.

Am I understanding your design theory correctly?