The typical 6-volt "golf cart" battery is rated for around 200Ah at a 10-hour rate. That puts you at a little over 1kWh, but you only generally want to discharge a FLA battery to about 50%, so call it 0.5kWh. You've got 2, so you're back to 1kWh total (usable).
I don't remember which 2kW inverter you have, but with it turned on, it's probably drawing about 30-50W. If your refrigerator is drawing on average about 50W, we can say you're drawing about 100W, which means you've got about 10 hours of runtime (not counting startup needs). It would be worth knowing what the refrigerator (and anything else on the inverter) is drawing--if it's around 150W total, that would put you right about 7 hours to 50% DoD.
But you're dealing with a slightly different problem, namely that the inverter isn't able to supply enough power to start the compressor. On mine, that spike is in excess of 1000W, only very briefly--if you're 50% discharged, that large draw may be momentarily dropping the supply voltage below the inverter's low voltage cutoff. A digital voltmeter likely won't be quick enough to pick it up, but a test light might.
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