Okay, I forgot that you wanted to be able to turn the inverter Fridge connection ON or OFF when the Grid is off, so here's the LTspice sim of a common latch circuit using one N-MOSFET, one P-MOSFET, one diode, and six passive parts, that is alternately turned on and off by a single NO push-button (S1) to drive the DPDT Fridge relay:
(The one in Post #20 can only be turned on, not off with a single PB).
It is energized by the NC contacts form the Grid powered relay, when the Grid goes off,
C2 insures that the circuit powers up with the output low (Fridge relay off and Fridge still connected to the Grid).
So when the circuit is energized by the Grid power loss (green trace), the Fridge relay can be turned ON and OFF (red trace) by the PB (yellow trace), where ON is Inverter power to the Fridge.
When Grid power returns, the 12V power is removed, and the Fridge Relay is deenergized, returning the Fridge to Grid power.
The MOSFETs can be just about any devices that are readily available to you.

(The one in Post #20 can only be turned on, not off with a single PB).
It is energized by the NC contacts form the Grid powered relay, when the Grid goes off,
C2 insures that the circuit powers up with the output low (Fridge relay off and Fridge still connected to the Grid).
So when the circuit is energized by the Grid power loss (green trace), the Fridge relay can be turned ON and OFF (red trace) by the PB (yellow trace), where ON is Inverter power to the Fridge.
When Grid power returns, the 12V power is removed, and the Fridge Relay is deenergized, returning the Fridge to Grid power.
The MOSFETs can be just about any devices that are readily available to you.

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