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Charging the solar generator after it is depleted is the hard part. Either plug it in so a 110 volt source which is typically at least an hour per hour of run time, use solar for many many hours per hour of run time, or run a generator for an hour at least per hour of run time. It is kinda of a zero sum game here.

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You are correct. These are hardly portable generators. But used within their limitations the can be quite handy…

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The battery/inverter theory is ok, but the problem is energy density/capacity of anything handy to carry. Nothing beats a gas generator for that. But one related cousin to the battery pack is to use an EV's enormous battery -- if you can use Vehicle-to-Load, you get 12...30...60.. even hundreds of Killlwatt Hours... which means about that many hours from a 1KW conditioner. Yes, you'll have to recharge your car if it's not a hybrid...

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And if you were able to run that AC unit for 1 to 4 hours, how much sunlight + time would you need to fully recharge that Jackery 2000 so that you could get another 1 to 4 hours of run time the next day? This is assuming, of course, that you are still drawing from that Jackery the rest of the day and night for your other needs.

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A 100-watt solar panel can provide 300Wh to 400Wh per day. So a 2,000Wh Solar Generator on a single 100-Watt solar panel would require at least 5 to 6 days to completely recharge. That’s assuming no other loads. I have 400-Watts of solar panels on my Jackery 2000 Explorer, and it will completely recharge in about 1 1/2 days with no other loads, assuming the sun is shining.

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What about the inrush current required to start the AC unit’s compressor? I assume you have a soft-start device installed and the compressor inrush current is less than the surge current rating of the power station’s inverter. If the power station’s surge rating is high enough to start the AC compressor, you could use the power station to start the AC and then keep it charged with a small generator (I.e., as long as the generator puts out more continuous power than the AC unit consumes.

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Some inverters can start an RV AC without a soft start device, I assume a Jackery 2000 Pro with 4,400 watts (36.6 amps) of peak inverter power should. Our 2000-watt Freedom X starts our Atwood AC easily and reliably. Our Honda EU200i would not on ECO throttle, so a soft starter was recently added for the Honda. However, I think your point is correct and we have done so, not with a large powerful suitcase solar generator, but with our own RV through the hot smokey nights of California fires. The Honda powered a True Charge 40 on ECO throttle to provide continuous charge into the battery bank as the inverter started and ran the AC from battery power. Why? So, a few dispersed others would not be disturbed by noise in the campground and we could sleep though the heat and smoke outdoors. Surprisingly the Honda ran nearly 8 hours before running out of fuel by morning. In the day, we ran the big noisy thirsty Onan.

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