I do NOT believe there's a way to operate the module by powering it b y battery voltage, and such an attempt will "let the smoke out", if in fact it IS still functioning.
Here's MY interpretation of how this system works, FWIW. Did you check AC voltage out of the stator with the trigger module unhooked from the stator? The way I understand this system, the stator makes AC, the trigger module rectifies it to DC, which is stored in the capacitor built into the coil.
When the tab on the flywheel passes the trigger, an SCR is "fired", which rapidly discharges the capacitor in the coil, causing current in the coil primary that is "stepped up" by the turn ratio to a high enough voltage to fire the spark plug. (A type of "CD" ignition.)
The open-circuit AC voltage out of the stator would be one thing, beyond that (whatever the correct value should be), the diode(s) in the module have to be OK to make the DC voltage, the SCR must not be shorted or leaky, and the coil and capacitor must be functioning so a DC voltage builds up between spark events so the coil is able to be fired.
So, if you are measuring DC voltage at the coil terminal and it's actually low, the fault could be in the stator, the trigger module, or the coil.
Are you aware most likely you CAN install a plunger and a set of points and a coil and convert it to Kettering ignition? (Seems to me certain blocks were not drilled for the plunger or point mounting holes. Normally, the holes are there and the plunger hole is plugged with a little "cup plug".
If you GGOGLE "Kohler 237750S" you will get info on what is involved.
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