The Honda will run the air conditioner ok, if batteries not down. I am probably mixing to issues together. I run the generator to charge batteries with the charger inbuilt in inverter, charges at about 100A. I also run generator to run air conditioner, but if batteries are down the load is to high for it. My setup is unique in as much as being in Australia we have 240V, so has a manual double trow treway switch, op 110V or generator, central of and down 240V true a step down transformer.
My question is the inverter has load sharing as if inverter can’t support the load it will take some power from batteries, how do this work, is it automatically done .