The lights could also be flickering because turning on a larger load causes the inverter to "hit" the battery hard for an instant until the inverter faults (which would indicate the inverter).
I would think that if the solenoid was bad it would cut out completely and cause the inverter to reset. However, if the solenoid is somehow compromised such that its providing some resistance (e.g. one of the connections melted or something causing a small contact surface area) a higher current draw would lower the voltage provided to the inverter causing that to wig out (although I would suspect you'd get some "low voltage" fault on the inverter in that case).
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