Just out of curiosity I looked up the 12V specs on the RMD8555 which seems to be a popular fridge. It draws 130W, so that is ~11 amps at 12V. So on paper with perfectly charged dual 6V batteries with 232 amp-hrs you would be drained in about 21 hours. In reality this would be much less because the battery is used by other things, batteries aren't perfectly charged and available amp-hrs are likely less than 232 depending on the rate battery is drained. Anyway this was just an exercise to get a ballpark on how long dual 6V batteries may last with fridge on 12V. Obviously you would have some coincident charging typically from tow vehicle or solar but the question is how much. I saw a post from John David from 8/2016 that said he did great with his fridge on 12V but he also had 320W of solar. That is double what ETI currently installs with their standard solar option.
While this is, as you stated, an exercise, I'd like to add a couple of points to make it a bit more realistic:
First, to avoid damage to your batteries, they should not be drawn down to less than 1/2 of their capacity, so that takes us down to 10.5 hours.
As to recharging, while the 320 watts of solar
might keep the batteries charged while running the refrigerator under ideal conditions, real world results will be a bit less. For example, I have 195 watts of solar on the roof & a 160 watt portable panel. On paper, the 355 watts of solar should give me almost 30 amps of charging current. Real world, I've never seen better than 20 amps of charging current from the combination.
Right now, with the portable panel aimed at where the sun would be if it wasn't for the clouds and light rain, combined with the roof top panels, my current charge rate is 1.1 amps with the solar controller in the bulk mode. Under clear skies I would be getting 15+ amps in the middle of a winter day, better during the summer, but with real weather, low angle winter sun, etc it is what it is. A couple of days like this in a row will leave you wanting.
Even with a pair of 6V batteries, I wouldn't count on being able to run the refrigerator on 12V for any serious length of time. If you must, I'd be more comfortable with doubling the battery capacity, and adding more solar or plan on using a generator to recharge the batteries. Propane makes far more sense for most users.