Why won’t my Champion C46540 generator run the microwave?

If the battery charger or hot water heater kicking on? Add that to lights, fridge and a few other minor loads and you could be overloading it...but I would expect a breaker to pop rather than the microwave to stop on it's own.

As mentioned, put a meter on the outlet adjacent to the microwave plug to see if you can see any voltage/hz issues. If you have a kill-o-watt device, put that in-line to see what the wattage draw looks like.