I wouldn't be surprised to find that your little EU2000 wouldn't run your 240v well pump if you rigged its 120v through a small step-up transformer. Simple engineering suggests that your generator should easily handle a one horsepower well pump. I have little idea of how large a pump you have but around here a 3/4HP unit feeds a whole cluster of houses.

The limitation is not power output but voltage. And a small transformer handles that. Being able to run your well pump would mean that if the mains went down you could spend an hour and fill the bathtubs, a kiddie pool or two, and/or a scad of five gallon buckets so the family has enough on hand to get them through a week. After a week you do it again.

It is also possible to keep the contents of freezers and refrigerators safe by running the units a few times a day. The better the unit is insulated, and the few times it is opened, the less often you need to run it.

Another thing to consider is that if you have something that needs to run that doesn't use much power you can use a good part of the excess capacity to run a battery charger to charge a battery bank. This battery bank can then be used to run light loads using an inverter. Running a few efficient lights, a laptop, even a small battery bank will suffice for many hours. The advantage is that your using capacity that would be lost running a tiny load on the generator. That and the battery bank and inverter is almost silent. Silence is very much appreciated if you have lived with a generator, even a quiet one like you have, for long.