Regarding the batteries, measure the boxes. If a group 27s will fit, use those. About the same cost and will give you more amp hours for dry camping. Otherwise use the group 24.
As for the generator, output voltage should always be 110 to 120 volts. It's the amperage/wattage that varies as the load/demand increases or decreases. If you have a volt ohm meter verify the generator output at both high and low rpm. You can plug in a hairdryer or some other load to verify the generator is providing the necessary power noticing the rpm increasing as the load is applied.
If you were trying to run the A/C on 15 amp shore power and the converter was trying to charge the batteries and it was during the day i.e. hot, then yes the circuit was overloaded and the 15 amp breaker did it's job protecting the house wiring. You need a 20 amp 110vac house circuit to run the A/C.
Keep us posted with your results
Oh, WELCOME to the forum!!