Ok, got it. Without seeing everything, maybe this will help. I do not understand what the relay is doing, so, I do not know what the control is doing to know where the inverter. So, you must figure out what activates the relay.
So, it could be that the inverter is fed by the breaker, and the battery, then goes to the relay. Or it could be the inverter is fed by the relay and battery and the inverter feeds your relay current output.
If you find the cable going to the refrigerator, the black should be coming from the relay and the white neutral from the neutral bar.
The only way that relay makes sense is if it makes the 16 volts dc and needs the neutral for the internal power supply.
Hopefully, someone that is a controls person will respond soon. I did power and didn’t l8ke controls.