Have been lucky enough to avoid using a generator so far. But... looking at my pretty similar 1-cylinder 4-stroke Suzuki outboard, I would say - no worries if left with gas for less than a month. No big difference with either wet or dry bowl, though I would drain it if it's a week or longer. For a month or longer periods of storage I run the carb dry and drain both the bowl and fuel tank. Stabil is added to the tank every time, it's a good thing to have with infrequently run engine. Without Stabil, gas starts deteriorating after a month.
If you're going to transport it - it's better to drain the bowl.