Generator ATS issue

Welcome to the world of non-standard residential grade generators and ATSs. The issue is as GoldDigger said, it depends on the proprietary design as opposed to the commercial world where everything is standardized with the smarts of power sensing always in the ATS and using 2 wire start.
If you have the ATS I suspect you have, it is a just a "dumb" switch in that it has no smarts in it and is just an electrically operated switch. That means you have to use it with the genset it was designed for as the power sensing is in the genset and the signal to operate the ATS comes from the genset. Problem is though that this signal to operate the ATS is usually proprietary and designed to work to work with certain manufacturer/model equipment.
Oddly, most of these low end, resi grade gensets (and I think the Kohler model you have) that do the power sensing in the generator can be converted to standard 2 wire start for use with a standard ATS. How ever, the inverse is not usually the case as in you situation-you cant use a proprietary ATS with a 2 wire start generator (in your case you would end up with no "smarts" to sense power outage) or a proprietary genset of a different brand.
All that said, there may be a kit from either Kohler or Generac to make this work. Some research will be in order. All this proprietary nonsense in the resi genset world would be laughable in the commercial world.

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