Mike, the question is not as crazy as it sounds. I have a friend who is finishing a 57 foot power cat which runs only on DC power. As I recollect, he has a 200 amp 24 volt DC alternator, running off a small diesel. He has 24 volt battery system (somewhere in the 2000 amp range)--each 2 volt cell is about 300 lbs.-12 of them to make the 24 volt bank. He runs two 3 KW 240 volt AC inverters to power AC, water maker etc. It is a very complex system--and expensive.
But to use a Honda EU 2000i there will be too much loss to do this. You will have loss between the AC generator, and the battery charger, the battery charager and battery and battery and inverter etc.
There is a 2400 and 2800 watt Yahama genset which puts out a bit more power than the Honda EU2000i. It may be enough to start some of the 11,000 or 13,500 BTU AC units.
Although you can buy AGM golf carts--that is not the reason to use them. Golf carts generally are good deep cycle batteries, with heavy plates. Yes, a single golf cart is about 70 lbs, vs 130 lbs for a quality 8 D battery.--and two golf carts are close to the power of a 8D. The Group 31's I use are high quality AGM's (I use the AGM because they are inside of the pilot house.). But not specifically deep cycle. They are mounted on end outboard of the aft dinette seat. One of the beauties of AGM batteries is that they can be mounted any way but upside down, and in the cabin. The charger is in the inboard part of that seat and the inverter is in the upper part of the hanging locker.
The prosine Tru charge is only a charger, not a combination. I have a separate inverter.
I think that a 15,000 BTU RV AC unit would be more than necessary for the C Dory Tom Cat or 25. I had a 15,000 BTU unit which cooled a 30 x 8.5 foot RV very well.
But again--probably best to size the generator which will run an AC on its own, and you can run this all night on the 6 gallon remote tank. Again I have not used the Yamaha 2400 or 2800.