I deleted my post last night because I had not looked before posting it.
The V-Twin Kohlers I have are 24HP Pro engines. They are factory equipped with oil coolers. These are vertical engines, if you are using the carburetor as the rear, the oil cooler is mounted on the right hand side towards the front of the fan shroud, there is a cut out in the shroud that allows the fan to blow through the cooler. So a cooler on these is nothing new.
And actually post #22 above mentions the same thing, and where the cooler connections are located at the oil filter base adapter.
And for post #33 just find a parts book and you have your cooler you desire.
Actually a quick check on Amazon shows them and the adapter together.