What other loads do you require?
Powering an Airconditrioner off batyteries is a by no means impossible, but is unlikely without gobs of battery and recharging power.
Alternator recharging is limited by the alternator to house battery circuit and its resistance, and the Vehicles voltage regulator, which basically controls the output of the alternator. Vehicles VRs rarely seek and hold a higher voltgae for very long and voltgae is electrical pressure, the higher the pressure the more amps flow, up to a point.
Basically relying ONLY on the alternator, well it is unlikely many hours of driving each day will return the battery anywhere near 100% state of charge, and for a lead acid battery to live a respectable lifespan it needs to get back to a true 100% every so often. The longer it goes without the 100% recharge, the longer it will then take to refill the battery to some percentage of its as new capacity. A lead acid battery is like a gas tank which keeps getting smaller the more time it spends less than full.
So employ any and all chartging sources whenever possible to get the battery to as high a state of charge as practical, or accept lesser battery longevity.
Cycling a battery to 50% state fo charge, with a high amp charging source early morning, can get the battery bank quickly to 80% state of charge, but 80% to 100% state of charge is likely going to take 3.5 more hours at the absolute minimum, and that is with the battery held at a mid 14 voltage for that 3.5+ hours.
Solar is great, when combined with a higher amp charging source early in the morning when batteries are most depleted and can accept the most 'bulk;' current from a high amp charging source, whether it be a well wired fast spinning alternator, or a high amp converter seeking 14.4+ volts powered either by thge grid or a generator capable of supporting the high amp plug in charging source.
