When you say it will run all day, is it mowing all day? The reason that I'm asking is that the amperage draw on the battery to operate the machine with the charging system not providing any output will drain a fully charged battery in about 2.5-3 hours.
There are two fuses near the control panel, one is for the machine electrical system the other is for the charging system, check them.
There is a wiring diagram in the operators manual.
As for the hour meter, they were problematic and have been changed in the Scag parts system to another vendor. I have had to replace the one on my 2008 Scag Tiger Cub.
Check all the electrical connectors on the machine, bear in mind that the charging system output passes thru the keyswitch and then into the electrical system, it is the single yellow wire.
As has been already mentioned, check the voltage at the battery with the engine running at half or higher throttle.
Mad Mackie in CT