the K181 has an optional fuel pump that operates off the camshaft. you can prob pick one up (or have it ordered) at a local lawn&garden place.
it mounts right under the carb, the newer fuel pumps have a plastic body, so be careful how you tighten it to the engine (and how far you tighten the fuel line fittings... i split one that way...

)
Kohler says the newer plastic pumps help to prevent vapor-lock due to the hot engine. which makes sense. i would try that out before trying an aftermarket thing.
you can get the owners and service manual from Kohler's site.
or i can try to get a pic of it and post it for you.... if i get around to it lol.
EDIT:
and the fuel pumps that were linked to operate off crankcase pressure pulses, so they are compatible with any engine. (might take a little work to hook it up though)