wrote
It might be a bad battery, but it's not an issue of it not
running enough to recharge it. These automatic systems
have a charger to keep the battery at full charge using line
AC power when the generator is not being used.
Presumably by now, days later
it's had plenty of time to recharge. He also indicated that
the battery charging indicator showed green, which
usually means the charger believes the battery is
fully charged.

Whatever it is, I agree it sounds like something between
the battery, starter solenoid, and starter.

Yep, I'd star by cleaning the battery terminals and the cable end on the
starter. A little rain a little corrosion, etc.