Provide Negative Trigger To Existing Relay

I put together this rough drawing to illustrate the setup. The large dotted area represents an existing circuit that receives a negative trigger from the BCM when the car is running. I need to supply a negative trigger to this circuit when the car is off. The two positive triggers control lights that operate regardless of whether the car is running. The idea is to use two relays to pass the negative trigger to the existing circuit. The only caveat is that, at any given time, all three circuits might be sending a ground signal. Does this get me close? Is there a better way to do this? Do I need some capacitors or diodes? Thanks Bill