Just an educated guess. The microwave is an "unfriendly" load for the generator, with its current draw not directly following the voltage waveform. The circuit breaker reacts to time-averaged current, while there might in reality be very short current spikes many times the breaker value that don't result in a trip. There could be a weak link anywhere in the generator circuitry that can't handle the short current spikes, followed by a protection mechanism detecting it and shutting the generator down. Could also be a problem with the voltage regulator being confused by the unfriendly load, too. Not knowing the Onan internals in any detail, I'd have to say the problem could be with almost anything in the generator part. My first shot in the dark would be to make sure all the electrical connections are tight, clean all of the connectors, and reseat plug-in circuit boards if there are any.
