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Does this unit plug in to a standard 120 volt outlet, such as the type we have in the US?
Based on my research, the unit draws approximately 5 amps. That being the case, I would look into a small generator to run the unit on.
Unfortunately, there isn't an efficient way to increase the capacity of an electrical supply to a structure. In other words, it's always substantially costly.
In addition, if the power to the home is unreliable, the generator is definitely the best way to go.
Although I'm sure you can make a number of solutions work, the two options you've proposed are outside of my realm of expertise. That being the case, it would probably be best for me to opt out of the question. This would give another expert, perhaps one with more experience using batteries for this purpose, to assist you. Please rate the expert who helps you to a suitable solution. Thank you.
You could install 2 large golf cart batteries, connect the 12volt DC output to a 1,000 KW rated inverter to supply 120 volts AC to the air conditioner, and then use the existing unreliable 120 volts AC to power a 1,000 KW rated battery charger to keep the batteries charged. That would work.
I am recommending the 1,000 KW rated items because they will not over heat as badly as 500KW rated inverter and charger which would theoretically support the AC, because any time you run electronics at their absolute maximum they over heat badly and do not last long at all.... additionally the 5 amp AC unit will pull close to 20 amps for 2 or 3 seconds when it starts and you need the inverter large enough to support that.
Net costs would be in the range of $600 to $1,000... the chinese items about half the price of USA manufactured items but with a generally much shorter service life ... a few months at times, not years.
If you care to rate my input so far positively, I will be able to afford discussing this a bit longer with you.