Fun post. I am glad that you are in an area where you can efficiently use solar. Without government subsidies in the area where we live they do not produce enough electricity to pay for themselves.
Our house is in an area near the foothills where the power goes out frequently. We and all of our neighbors have generators ready to go whenever their is an interuption. Fortunately we have natural gas as well. When the power goes out all the generators start up almost immediately.
We have ours metered so that we know the exact amount of power and gas that has been used. A little secret... unless you have something that uses a lot of power or creates a large starting load an over-sized generator does nothing but waste fuel as compared to an appropriately sized generator. Generators are most efficient near their peak capacity. When the generators are outputting 2500 watts the 10kw will be using almost twice as much fuel as a 5kw.
We use a 5kw and try not to use two heavy loads at the same time. Since we have gas heat and use a gas clothes dryer this basically means that we try not to use the microwave and toaster at the same time.
I believe that's not an issue in my case. If I understand it right, if I have my solar inverter in off-grid mode and it decides it needs generator power (i.e. because my batteries' strength of charge SOC reaches a set point like 20%), then when my inverter fires up the generator it tells the generator to run at full capacity until my battery SOC is up to a set level (i.e. 30%) and tells it to shut off. So if I have 60 kWh of storage, 10% of that is 6 kWh. That's about 40 minutes of run from a 10kW generator. Maybe 60 minutes to account for my home at night consuming a little power. We're talking less than a gallon of ethanol free gasoline (to keep it from gumming up the carburetor of the generator if it sits there for a while) per month.
Based on the output I've received from my solar system so far, I believe I could double the system and need about 45 minutes to an hour once per month from my generator, to supplement my solar to power my all-electric (once I replace the gas appliances) two story home as well as throw in 15 kWh daily into an EV (that I plan to buy next year whether or not the power company forces me to go off grid). The reason I want two 10kW generators if I go off grid is because I every now and then have family in for vacation (consuming more power than just my wife and me) and because I'd like a 2nd generator as a backup in case one quits.
But that's only if the power company forces me into either going off grid or paying their ridiculous monthly solar fee (that I'm so far avoiding). For now I'm happy with it as it is.