What I have found with both my small Honda Generator, as well as my 2 Westinghouse 9,500 Watt models, is that while they are all air cooled and splash oiled, they warm up very fast. They also have very small crankcase capacities. (Not much over 1 quart).
If I start them on say a 45 F morning, the crankcase metal is warm to the touch in just a minute or 2. In 5 minutes you can't keep your hand on it. So if you do use a heavier weight oil, (I use Mobil 1, 15W-50), it will thin out quickly.
And when these things are running under a heavy load, they can get very hot. (Both the engine as well as the dynamo). And they usually run for hours on end. In hurricane country it can be for days, or even weeks.
So leaning to the thicker side of oil viscosity is going to be more beneficial. As opposed to a thermostatic controlled, liquid cooled automotive engine, with a 7 quart crankcase, that only gets driven on short trips in Duluth in December.