Wayne: My amp light once came on with my 782 equipped with a Kohler Mag 18 motor the same way you describe; it turned out that the magnets had come loose inside the flywheel so the "alternator" wasn't functioning properly. The engine ran fine off the magneto, it didn't need the battery; but nothing else worked right, including the lights. Your mowing issue could be due to the PTO not getting enough current to hold itself in properly.
So, I suggest checking the output of the alternator under load, that is, do full current test, not just voltage.
I suppose a bad voltage regulator would also cause charging, and/or running problems.
Finally, have the battery's health checked too; it may not be able to store sufficient charge.
Just some thoughts for you to consider, hope you get it sorted out.