adding a start capacitor to a circuit

[1sharpguy]

Attached is a wiring diagram for a Kenmore compact refrigerator. For reasons I will explain below, I was wondering if I can install a start capacitor between the thermostat and the overload protector in order to assist in starting the refrigerator compressor's motor?

I am powering the refrigerator with a 12V DC/120VAC inverter connected to three deep cycle 12V 109AH batteries. The inverter has a low voltage protection switch. The cutoff is 10.5V. When there is a power discharge and the voltage as "seen" by the inverter drops below 10.5V the inverter shuts down. This invariably occurs when the refrigerator cycles on. This is happening because the refrigerator does not have a start capacitor, but rather a PTC switch which drops the refrigerator compressor motor start windings out of the circuit once the refrigerator is up and running. The initial surge in power (current) required to start the compressor is causing a low voltage fault in the inverter shutting down the system. Since at this time, I would prefer not to mess with disconnecting the low voltage switch on the inverter, my solution to the problem is to add a start capacitor to the refrigerator circuit.

Bringing us back to my question above, would it make sense (be possible) to connect a capacitor between the thermostat and overprotection switch in order to start the compressor without having it result in a voltage spike "seen" by my inverter. My thinking is that the capacitor will give the needed boost in power to start the motor running and then be dropped out by the PTC switch. I don't know enough about capacitors or circuitry or the other components to say whether this will work or not and solve my problem. Was wondering if someone on this forum might be able to assist. If this is doable, what microfarad cap would your recommend? The compressor motor is LRA 6.6A and RLA 1.3A. The voltage is 115VAC. Additionally, I have another concern with this set up. Will placing the capacitor in front of the Voltage Overload Protection in the circuit enable that feature and shut down the compressor motor as well?

Another solution that has been recommended is to replace the factory PTC and overload protection switch on the compressor with a Supco 3 in 1 relay, start capacitor and overload. The only issue I have with that is the relay is a potential relay. At a preset voltage surge, the relay opens taking the start capacitor out of the circuit. Thus, not sure how lower voltage from discharging batteries would affect its operation. Unfortunately, this is where I fall down understanding how the AC voltage coming from an inverter is affected by the dicharging of the battery bank with time. I am presuming there will be a voltage drop on the AC side of the inverter. If the potential relay does not "see" a spike in voltage then it will not remove the start capacitor from the windings and thus run the risk of damaging the motor windings over time.