The generator rpm should remain constant, as the load changes, to maintain the 60Hz (US) or 50Hz (ROW). What will have an audible difference is the throttle / power automatically changing to deliver more / less output as the load changes.
To state the obvious, the breaker is tripping because excessive current is being drawn. That abnormal condition should not happen with the factory-installed devices.
A good place to start would be make a concurrent voltage measurement at the load while the load changes. The easy method would be to insert DVM probes into the spare NEMA 5-15 receptacle at the microwave and measure the voltage when idle and when the microwave is operating.
Those measurements should be the same, within about 5VAC. If not, then one potential issue is excessive voltage drops in the system. If not the same, then perform those same measurements within the generator, where the stator connects to the internal circuit breaker.
If the stator voltage remains constant, but the voltage droops at the point of load, then part of the distribution is bad and should be fixed. Move away from the stator, such as the other side of the internal generator breaker, then to the salon distribution system to discover where the voltage drops exist.
Note that when the voltage droops, the load is relatively constant, so the current must increase to satisfy the load. When that current gets too high it will trip the breaker.
Another option is that the breaker has gone bad. Use an ammeter to measure the load current to verify the current is normal and the breaker is tripping.