Believe it or not they make special square holed wrenches specifically for removing sump plugs and they are sold at most auto stores.
Secondly you had your maths right the first time a 12 pointed ring will fit, but it has to be a good quality one and it has to go right up hard against the threaded region or it will do exactly s you feared.
4 does not divide into 6 so a flat side drive ( impact ) will not fit.
1/2" is correct an open ender will do fine, the drain plug is not done up all that tight.
In the long run you will be much happier with an extended drain hose with a spring loaded bayonette fitting on the end so you cad forget about wrenches after it is installed and forget about burning your fingers on the hot engine or covering the engine with hot oil.
There are a couple of types but the ones with the valve on the free end are the best
The ones with a cap are the easiest to use but you really need to have the draining end clipped tight against the mower and higher than the level of the sump
The one Kohler fitted as standard with the valve near the plug can be awkward to get at.
In the end I used a Gator Grip Universal Socket on the original plug, and that worked fine.
It was a pain to drain oil with the drain "chute" that came with the mower. Be prepared for a mess. I can't imagine going through that every oil change. :thumbdown:
If I had to do over I would pump/siphon the oil out instead and then switch the oil plug out.
Should be all good moving forward with this Drainzit Oil Changing Drain Port.
The drain plug on my lawnmower is in a tight spot. The 1/2" drive end of a 1/2" x 3/8" ratchet-socket adapter fit nicely on the 1/2" square drain plug. Then I used a 3/8" socket extension bar on the 1/2" adapter to extend it out beyond the engine and a pair of vice grips to grab the extension bar. Was able to remove plug without rounding corners.