Noting that Ethanol-flavored gasoline and yard equipment don't get along together, reputedly shortening the life of your mower to as little as 3 years, and also noting the complaint that some car drivers have that if the water content in their gas tanks gets over some critical mass, the ethanol and water simply fall out, there may be a solution to problem #1 here.
Ethanol in gas, while reducing the energy content of the fuel, does have the virtue of removing condensate water from the tank as it will mix with the alcohol and be passed through your engine. One would think that deliberately adding water to the gallon of E10 you bought for your mower and shaking it vigorously until the water and alcohol come back out of solution might be a way to remove at least part of the alcohol from the fuel.
The alcohol-laden water will sink to the bottom of the container, and the remaining fuel can be poured or siphoned off into another container. A gallon of gas and 2 quarts of water in a 2-gallon container seems like a good starting point. Any takers?
My mower is approaching 10 years old and still runs fine, so I have no incentive to try this. FWIW I also make it a point to run the tank dry at the end of mowing season. It makes for easy starting the next year.