A fully charged battery is 12.6V.
* pen light batteries will produce 12V, but not have the capacity to crank an engine, so volts by themselves don't mean that much.

I'd suggest checking all cranking circuit connections for high resistance. This includes the ground circuit too, such as tight engine mounting bolts etc.
New batteries are bad much more frequently than one would expect.

Why would you replace the stator?
Did you check the output? How about the Voltage Regulator? What voltage are you getting at the battery when running at high speed?

Charge up the battery to at least eliminate that as a variable

Check the compression, if it is HIGH you may want to check the valve adjustment gap to make sure they are opening at the right time and not late. High compression requires more amperage to turn the crank, hence the jumper cables. I had a similar problem with B&S intek 23HP, very slow cranking; neighbor had same exact problem same exact engine no crankiing. If you do not have a compression tester remove the plugs and see if it cranks easily, if so adjust the gaps ASAP.

Which might apply if he had a Briggs instead of a Kohler.

Apparently the OP doesn't want to respond, so it's a moot point.