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As far as the A/C not working under generator power, may be that the generator isn't sized big enough, or to many other appliances were running......, need at least a 2400 watt genny to run a 13.5K A/C.
As mentioned, assume you have power at the pool house receptacle. Did you check the 30amp in-line fuse next to the battery? Loose wire at the converter/charger?