I'm struggling to understand how this works-out in practice over a period of many days with only one 360W (~3000Wh/day) PV panel to recharge those batteries.
It doesn't. Running the AC on batteries is useful on those one night stops at a winery or some other place that doesn't have electric hookups, typically on the way to a place that has hookups.
When I'm not running the AC (e.g. skiing trips in the winter, fall camping, etc. ), the batteries provide me power to run other things like the convection oven, plumbing heat (to prevent freezes), laptops, etc.
Think of this as the electric version of your fresh water tank.
Edit: I realize the confusion was that I said i could run my AC for 24 hours on a charge, which makes it sound like I can run my AC 24 hours a day, ad infinitum.
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