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Hmm. Energy capacity of 240Ah battery approx 240x12/1000 = 2.8kWh, allowing complete discharge in not less than 20 hours when new. Used cyclically at 5-10 hour discharge rate, as battery ages, don't expect more than 2kWh. Icemaker consumption 240W / 90% inverter efficiency = 270W. Run time = 2000/270 = 7.4 hours. Is that what you meant by a long time?
When looking at refrigeration energy usage in kWh/year, that is averaged over summer and winter for a specific amount of door-opening and product chilling. For a domestic fridge, opened ten times a day with 2kg of produce chilled, the consumption might be low as you suggest. But in peak summer, with commercial usage, you might have the compressor running all the time. Nothing is actually 19W, that's just an average. If the compressor is running it's whatever the compressor power is, a few hundred watts for a larger unit as you suggest. If you had four refrigeration units and a total of 1kW running load, your 240Ah battery would be caned by the 100A discharge rate and probably run you for a couple of hours max. unless everything was in your favour (new batt, cool weather, all produce cool when loaded etc.)
Also take heed of Marvo's point about optimistic figures. I was surprised by the 240W rating too, they probably aren't lying but everything needs to be just right to achieve the quoted performance. Feed it warm water or stop it overnight or whatever and it probably won't stay in spec.
