Lights, Electronics, pressure pump, starter all run off of 12VDC (Battery) Fridge is typically dual-voltage (12VDC/120(110)VAC) . AC, Heat and Battery Charger are 120VAC only. So the AC will never run off of the batteries regardless of whether you have the boat plugged into shore power or not. To run the AC off of a Genny, you will need 20A male to 30A Female converter and a shorepower cord to connect the Genny to the shorepower inlet.
I use a Honda 2K to run a 7000 BTU genny. A marine genny is nice and I would prefer it. Plenty of people run portable generators. You just have to be careful about Co2. CO2 is heavier than air, so if its on the bow the hatches and portholes need to be closed. Particularly those below and downwind. You'll also want to close all the isinglass to prevent gas from running through the cockpit. It's easier but a little louder out on the swim platform.
For those worried about CO2:
I place mine just in front of the windshield, turned sideways with the exhaust pointed outboard and just slightly aft. I anchor off of the bow only, so the bow is pointing into the wind. I close all my isinglass along with the hatch beside the genny and the porthole below it. I leave the portholes and hatches in front of the genny open to allow air to move through the cabin. I have a CO2 detector in my cabin and it doesn't go off running it this way. I'll run it for 2 to 4 hours per day with no issues.