I just checked out transformers, which is what you would need to run off of 110 volt in North America. They are about $40 to $50. I like Blars' idea of using a European inverter, but inverters are expensive. Transformers are one of the most efficient machines made at about 98% efficiency. That would use about 7 watts or so on top of the 350 watts, which is very good. Yes, the AC is unit is expensive, but having extra batteries or solar or generator to operate a larger AC unit costs money, too. Having an AC unit at less than 400 watts is great. It might not make your place cool, but it would certainly take the edge off. I would certainly appreciate that during the brutal summer heat; a guy will do almost anything to cool down when it's 100+ degrees, so the cost is relative. Or if you have pets that must have cooler temps.
The smallish size also helps; every square inch matters. It also isn't roof mounted, so no clearance or leaking/pooling issues. And it doesn't take up roof space for solar.
I bet damaging the umbilical cord would be easy. Looks fragile.
Ideally, you would want to be able to run it at night as well, but that would require a larger battery bank. But if you insulated a small space well, it might run only periodically at night, so the amount of energy needed to take the edge off might not be as much as one would think. You could drape a blanket over your bed (like making a fort) and just cool that little area or somehow block off a smaller sleeping area. If it used a total of 500 watts over a 10 hour night, that would totally be doable. At the least it could run full blast during the day when you needed it the most.
I like it overall. Thanks for the info. Lots of us have scoured the Internet but haven't found this unit.