That piston is junk. I would rather give a better evaluation after seeing the bore and more views of the piston but...
You have some pretty good scoring. The top of the piston looks like there was carbon buildup it was slamming into. Carbon buildup can make the engine run poorly and from the looks of the piston, oil was getting past the rings.
How much carbon was in the combustion chamber? I'm wondering what the valves look like in this engine. Perhaps this engine could live again. You will know once you measure the bore with your new tools.
Head warpage on these engines is repairable as long as they haven't been machined too many times. It can normally be repaired with a flat surface, sand paper, and some time if you don't have access to a mill.
I guess how I got to this point got lost in the previous thread perhaps. The scored piston is out of the old block, which had been bored to 030 and I found it with the head gasket broken. Perhaps pieces of the broken gasket are the reason for the scored piston. It was smoking badly and running really rough. There wasn't that much carbon buildup. When I posted at the time, the combined wisdom here concluded that most likely the rings weren't installed properly by the previous owner. So I bought new rings and was going to just give the old block a hone and new rings. However, since the old block is at 030 already, and at the time measured at 3.533 roughly, and having come across a STD block on ebay, I just decided to get that block and do a full rebuild, so here I am. Got the STD block which measures quite good, and will get new piston and rod, etc.
I'll still check the old block for scoring, just because I'm really curious.
I was aware that piston came out of your old motor. I like investigating failures too. I'm just wondering if measuring with your new telescoping gages would show that this block is still useable. Hone and a new piston and rings doesn't cost much if the block is OK. A 0.030" overbore will give you more power. Then keep it as a backup.
Some of the Kohler K parts are getting hard to find. Cylinder heads, blocks and cranks are getting rare and sometimes expensive. I had to tear one of my parts engines completely apart to get a lifter for one I was rebuilding.