As the OP stated the problem, it isn't practical. But running one or maybe two small A/Cs overnight on a big cat may be practical.
A 5,000 BTU unit will cool a state room in the evening. In fact it will probably only have a 50% duty cycle after initial cool down. A 5,000 BTU unit draws about 6 amps AC. That is about 70 amps DC input to the inverter
and at a 50% duty cycle it will take about 300 amp hours to run it all night. Not too bad a load for 3-4 pairs of golf cart batteries at 660-880 amphour capacity.
Double those numbers for two staterooms.
But what comes out must go back in. It will probably take at least 3 hours of genset running time to put back those 300 amp hours as the max you should charge flooded cells is 50% of their capacity and the charge rate will drop as they get full. And you need to run the genset at least an hour before bed
to cool things down first.
But it is a good way to keep a genset well loaded, running it 4 hours each day to recharge the batteries and cool down the stateroom. Many gensets run all night long at a very light load and end up with glazed cylinder walls.
So, if all you want A/C is for comfortable sleeping at night, this might work. I have read of a few who have tried it.
As far as a packaged solution Mastervolt makes a "hybrid genset" which packages a small diesel
generator with an integrated inverter/charger. The genset automatically starts and stops to supplement the inverter and charge up the batteries. So far it is only available in 220/50Hz though.