Bob,
I was going to suggest essentially the same thing as dtchild1. The question becomes how to test it--testing the batteries would identify a bad battery, but wouldn't be conclusive in showing that voltage isn't dropping when the fridge tries starting.
I would try connecting a test light (or ideally an analog voltmeter) at the inverter's input terminals. A digital meter (at least typical consumer grade) isn't going to show a momentary drop. Depending on how hard it is to get to the stuff, I could also see testing with a regular starting battery in parallel with what you already have just to help narrow down what's going on. I'm still a little surprised that the 600W inverter couldn't do it, but the compressor does briefly demand quite a bit--but for a short enough time that it wouldn't trip a thermal circuit breaker.
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