With that said, the Inverter is probably blown. The question becomes 'why?', and the changes which should be made for future reliability will likely be costly.
The only inexpensive cause for the experiences you described would be loosened wire connections or a partly failed circuit breaker. I have seen a couple of cheap circuit breakers fail with an inability to pass large amounts of current high current, while still showing good voltage across the terminals while "on". Check all battery bus bar connector nuts, and the nuts for the wire into the Inverter bolts. If they are all OK, then it's going to be a costly fix.
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When DC -> AC Inverters fail over a period of time (rather than "immediately"), it is typically a case of DC Voltage starvation or highly reactive AC loads - or both. The 3000/6000 Inverter which we both purchased (
https://www.ebay.com/itm/335060051180 works great for non-reactive loads, and I have not yet hurt mine in an obvious ways by also using for a microwave rated at "900 watts" for very limited periods of time.
But your 120-VAC only refrigerator may be far more "unfriendly" (reactive) than my microwave. My air conditioner loads are greatly smoothed out by a complex and expensive "start-up" device, which also overrides the "run" behavior: A MicroAire easy start, IMO superior to the cheaper "Soft Start RV" and similar units (
https://www.ebay.com/itm/202724244901).
So even though I run the older Coleman Air Conditioner pretty frequently, my 120-VAC loads are very likely less damaqing than yours.
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You most almost certainly address the DC side as well. Your "150Ah" battery is less than 1/3 the size of my own battery bank, inherently capable of less current from the cells (even with a great BMS and AWG 4/0 wires into the inverter.) The 6000 watt peak rating corresponds to input power approaching 600 amps, and every bit which can not be supplied from the batteries and BMS becomes a destabilizing drain on the capacitors and driving circuits within the Inverter.
Over-stressed to many times, the capacitors probably started to fail - leading to a quick cascade of additional failures. You probably need both a bigger inverter AND another battery to run this fridge. Maybe you should buy a different fridge instead?