I wish you luck, and a whole lot of it.
Tip for working on Kohler stuff. OEM, OEM, OEM. It is nearly impossible to find any aftermarket parts that work correctly on their stuff.
The older, 3.5 gallon Rialtos were bad enough. That modified fluidmaster in the ones you show is a pain and a half to get parts for.
The Kohler rebuild kit for the old Rialtos runs ~$80 cost for a ballcock and flushvalve. For the ones that you show, I don't know the cost. I do know that Ferguson had me call a Kohler rep to try to get a kit. Wait time was a week or two. The customer decided to go with a new toilet instead.
The problem with that particular toilet was a leak on the 'sheath', the white plastic tube that surrounds the water inlet to the ballcock. A leak there will appear as a drip from the supply tube.