How good are you with electricickery ?
Take the blower housing off both motors and follow the various parts back to the engine plug.
If necessary plug your old engine back into the mower because often the wires change colours at a molex plug ( cause it only goes in one way ) between the engine & the mower,
Take a photo or mark the wires on the mower side, coil, alternator, carb etc.
The Kohler will have both a + & - to the carb while the briggs the - will be a short wire bolted to the engine cover.
If you are still confused then take some nice clear in focus photos of both and post them here.
Only one vital thing to worry about is the magneto kill wire.
Remove it from the magneto when you think you are done and check it with the key in all positions and all of the controls in different positions, brake on,brake off etc.
This is a ground wire and must never ever see battery voltage for the slightest time or you will fry the chip inside the coil & it will be toast.
It should be open circuit ( infinate resistance or OL ohms ) in all running positions and short circuit ( 0 ohms ) with the switch turned off & with the switch on and no one in the seat brake off and the same on operator & blades on .