UPDATE
After mowing all summer with no issues, the same problem has cropped back up. The first time was while I was mowing recently, and I noticed a LOW BATT message on the LCD as the engine started to sputter. Barely made it back up to the shop before it died. Battery was at around 12.3V. Putting it on a 4 amp charger for a while brought it back up, and it started okay at that point.
Took it to a tractor show along with my 8N, and it died again while driving it down to enter in the slow race, and would not start (not turning over, not even a click from the starter solenoid). Battery was at 12.4V on the LCD screen. Let it sit a couple hours and it started up, which allowed me to get it loaded on the trailer. Would not start when I got home, however. Put it on the charger a bit and it started, so I unloaded and left it in the shop.
Put it on the charger overnight, and when I checked the next morning, it was back to "no start" condition (no click, not turning over), but other electrical equipment worked fine (lights, deck motor, etc.). Battery indicated 12.8V, engine off.
I pulled the engine-to-body wiring harness connector apart and cleaned those contacts up (there was a bit of white 'haze' on them), and put a squirt of OxGard on the pins. The Kohler CH740 service manual has a test that says to depin the white wire from this connector (safety switch ground) and attempt to start to determine if a safety switch is the issue. Tractor started right up, so I reinstalled that wire in the connector and the tractor started again without issue.
I mowed the grass later in the day and the tractor ran fine the whole time.
After I was done mowing, I let it cool down, then ran the tests in the service manual for the voltage regulator/rectifier, to see if I have an insufficient charging issue due to observation of the battery voltage repeatedly getting low. I initially couldn't even find the rectifier (!), but after searching pictures of the CH740 online, saw that it's mounted on the side, down low at the flywheel end of the engine. As installed in the XT3, it's located on the engine such that it's mostly hidden by the tractor frame, and I ended up removing the black plastic filler panel below the rear of the hood (left side, facing forward) to gain better access. All tests passed without issue, but the connector itself had a lot of crud in it, so I scraped it out as best I could and put a shot of Oxgard into each pin, then put it back together.
After sitting overnight, the battery was reading 12.6V, but it started and ran fine so I moved it back into the garden shed. After shutdown, the battery read 13.4V on the readout, but was dropping - same thing I saw with it in the shop. Don't know if that's a clue or not, but thought I'd include it.
Appreciate any suggestions based on the info provided.
es