Do you have 12 volts at the Battery wire on the voltage regulator. Is the voltage regulator properly grounded. May need to make sure the battery wire to the regulator can carry current. I have seen 12 volts at that wire but won't carry a charge current.
Sounds like it's time to pull the flywheel and have a look at the alternator under it and make sure all the wires are intact. You also need to check the magnets attached to the inside of the flywheel to makes sure none have come loose. Don't really matter what engine brand. they are all basically the same if it's a V twin.
Billy had replaced a battery and two voltage regulators on his '90s 25 hp Kohler. (Country Clipper) It still wouldn't charge. A charge wire from under the fan cover to the battery was pinched under vs fed through a gap as intended. Several years of mowing had allowed the insulation to split and ground out the charge. (>25v AC IIRC) His alt/gen coils weren't harmed at all and he now has $150 worth of spare parts. (2)
When I took auto mechanics in college my best instructors taught that w/o proper diagnosis a guy won't make a living fixing what he doesn't figure out first. I like to think the OP is OK and busy operating vs checking in.