Probably poor wiring and/or shielding of the pickups. I mean, you have definitely pinpointed that it is the specific guitar that is the problem, not your KPA. I would re-solder the pickups. If that doesn't work, I would replace the pickups. At the same time, you should look at shielding the pickup cavity...assuming you want to stick with true single-coil pups. Personally, I would toss them, and go with something like Dimarzio HS-3s.
I can't help but thinking these are not causes of the issue since the effect is only present on the hard hits.