I am not familiar with the Koehler wiring. I am going to assume it is like my Kawasaki and the osha switches allow a hot wire from each switch to the module.
So I would learn which wires on the module come from where. So next time it does not start you could check the module with a test light to see which switch is not sending the power to the module.
I had this same type of issue. I bought a $100 module and then saw a problem with the connector while changing the modules. I bypassed the connector on just that one wire with a jumper wire. Then just used the old module and the mower has not acted up in a year. I have the new module in my parts drawer. Good luck.