One problem with getting a 230V inverter is that they are mostly designed to operate in Europe, with 50Hz frequency. You would also need a 50Hz air-conditioner to go with it. And, you still have a regular 120V 60Hz inverter operating right next to it.
I think a better option would be to get a split-phase 120/240V inverter that natively runs at 60Hz. That way you only have one box supplying both your 120 and 240V AC. I have this one.
Free Shipping! Schneider Electric Conext SW4024-120/240 Inverter/Charger, 4000W, 120/240VAC, 50/60Hz, 30A Transfer, 24VDC, 90A Charger, RNW8654024
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Another advantage of this inverter is that it already has a built-in charger function. You simply connect generator power to the ACin terminials, and the charger converts the generator's AC into battery charging DC.
So, this one box would replace three of your installation items.
BTW, Schneider also makes a 48V version of the same unit.