During spring and fall the temp will swing quite a bit here. There is many times I have wiped frost off a boat seat and had it 85 by the afternoon. I do plan and using a generator for heating also. My 5th wheel heater seems to be a battery killer.
There is even a picture of an EF2400iS generator on the label. Interestingly the bottle says 1 US quart, which means this is not a locally produced product, as Australia uses metric; the bottles sold in Australia are typically 1 litre packs.
It can and does get pretty hot in summer in Australia, easily 100 deg F at times, so presumably these engines can handle the heat with SAE 30 oil, so I don't expect oil shearing to be an issue with this weight oil. The only real way of knowing for sure is to get the used, 100 hour old oil analysed.
Putting a "heavier" oil will probably increase the fuel consumption of the generator and may not provide any extra protection than what the 10W-30 oil is already doing.