Agree with Star on this
The kill wire links the two coils together
When the coils fire they send a ripple down this wire which cause to other to fire at the wrong time
As Star mentioned there is a diode inside the coils to prevent this happening
Yours has failed open circuit
Because of that you could try replacing the kill wire with a B & S one where the diodes are in the wires not the coil itself
Or just add a pair of diodes which are about 10¢ each .
I have never done this myself because I have never had your problem present itself
Due to changes in the Kohler modules over the last few years, basically you need to change both modules, partially because in some cases replacing one module and the other will fail shortly, and with the redesign of the Kohler modules if the part numbers on the module don't match they both need to be replaced due to the new modules have a high voltage kill system compared to the older modules were low voltage. The high voltage module will kill the low voltage module