First, your alternator is large enough to work if you run the engine at a high enough rpm--maybe 2,000. A throttle lock would be nice. A quality deep-cyce can absorb about 40amps of current per 100AH. A pair of group 24s (about 65AH) would be easily serviced by your alternator drawing 50amps max. during charging.
Keep in mind that a cheap high-amp alternator is a piece of shit for doing what you are doing. I would not swap out a stock one for one of those. A marine alternator, like a Balmar, is meant for that. They don't sell one over about 130amps in the standard frame sizes that would be found in a car or truck. A 200amp Balmar would weigh 25# and cost a couple thousand.
Next thing is, it's a bad idea to discharge a deep-cycle battery below 50%, and a bad idea to discharge a starting battery below 10%. So for a deep cycle you need about twice the bank size that a calculation would give you. For instance, if you want to run a 5,000BTU window unit that draws about (a guess) 700watts while it's running, that would drain completely a single Group 24. So you would need 2 to keep one from going below 50%.
Another thing to consider is the inverter. An inverter isn't good with surge loads. A small 5,000BTU window unit runs at about 6amps (a guess) and whenever the compressor starts probably surges to 40amps. An inverter doesn't have a long surge capability like a generator--which can lug along for a while before popping it's breaker. I would get a huge, cheap inverter and hope for the best. Definately web search and see what exact combinations people have had success with. I decided against risking this with my bus conversion because a couple thousand bucks worth of batteries and inverter and maybe it would or maybe it wouldn't be able to start the AC. I ran a 5,000BTU off of my Yamaha EF1000--which is a very small generator. I would lug like hell then finally the AC would hum to life. I kept the AC on full blast and used the camper windows to keep it warm enough to not allow it to turn back off. You might consider 2,000Watt Honda or Yamaha instead of batteries. They are quiet, reliable, last more than 200 cycles (which is the expected life of a deep cycle flooded battery) and there is no risk in it not working.