I had a similar problem with a 2005 with a Generac engine. The charging circuit worked fine while the engine was running and produced sufficient voltage into the voltage regulator. The regulator functioned properly producing 13 - 14 volts at the output to the battery.
However, when the engine was shut off and the ignition turned OFF the battery discharged within a short period of time to about 10 volts.
After a thorough testing of all other parts I concluded that the problem was some sort of defect or short in the stator. Rather than replace the stator (since it was producing proper charging voltages when the engine was running) I decided to try something else.
I cut the battery charging line (between the voltage regulator OUTPUT and the battery terminal on the starter solenoid. This stopped the battery discharge immediately and the battery remained fully charged for several weeks. I then installed a 12 volt single pole relay (NORMALLY OPEN) into the charging line and wired the relay coil to the ignition switch of the mower so that the relay would only activate and close when the ignition was turned ON. This will allow the battery to charge when the engine is running but WILL NOT allow any backfeed discharge into the charging circuit or the stator.
WORKED LIKE A CHARM!!!!