I would suspect a the generator is slowing too much and the frequency is dropping too far below 60 Hz. It could also be dropping the voltage too much. Have you measured the frequency or voltage when it's running? I'd compare it to when it's running the coffee maker as that should be similar sized load. Coffee makers probably don't care much about voltage or frequency.
Keep in mind too, if your microwave is rated at 1,300 watts that's usually it's "cooking power". It's actual power draw from the generator will be a couple hundred watts more. That's not enough to make a difference unless you're running other loads at the same time maybe.
