Yep. The 1B1X should do it, and as I found out, there are a bunch of inexpensive versions of that valve out there. Easy to replace. Works great. The one I bought cost $9. You may have to reuse the old ball and rod, because the compact ball and rod in your toilet aren't widely-available.
Now, if you really want to have fun, you can try what I did with my old American Standard lowboy toilet, that is a little like your Kohler. After my successful experience with the $9 1B1x, I did a little experimentation. I tried installing a Korky 528PRO fill valve, just because. You can get the 528MP Platinum valve for like $12-13 at most hardware stores, which is equivalent. I just ran the little thin hose into the bigger hole in the toilet for bowl refill, which worked perfectly. The valve doesn't go as low as the 1B1X does, but on the lowest setting, the water jussssst shut off when the water level reached the overflow. Since put I in the Korky: (1) the toilet refills very quietly; (2) it takes a little longer to refill because it is filling higher and the 528PRO doesn't have quite the water flow that the 1B1X does, since the latter was made for water-hogs; (3) it shuts off smoothly and, for the first time since I moved in more than a decade ago, the toilet works perfectly, or at least as well as it ever has. Turns out the higher water level in the tank helps.
You should have a look at your toilet to make sure this would work. So, you do the experiment at your own risk, but if you want to check out the thread with my experience in it, it's here:
https://terrylove.com/forums/index....alve-on-as-galleria-toilet.65751/#post-488686
You can ignore the lengthy descriptions and just look at the photos.
It might make your toilet run better... Or, it might not. Either way, we would all appreciate your description of what you do and how it turns out, preferably with photos.