Yes, your converter is just another load and will be labelled on the breaker box. If you look at it, many converters are capable of putting out 45 amps at 12 volts. That works out to 540 watts if it's working hard trying to charge low batteries. Not often it would be running that busily, but it is a sizable sum.
Would it help to have as much load on the generator as you can, then suddenly switch it all off, and turn on the air conditioner before the generator has a chance to throttle down? I know, if I have my genny runny on eco mode and then add a big load it stumbles. But if I have it running on normal, I hardly notice the difference. Try adding a small heater just before you switch.
