This is not a known issue, and it’s the first report I’ve heard of this, so it would appear to be isolated issue on your machine.
I am not able to offer an immediate solution, but here are my thoughts/questions:
What version of PopClip are you running?
Have you been using PopClip for a long time, and has this recently started occurring, or did you only recently install PopClip and it’s happened from the start?
You said it happens when PopClip is installed. Do you mean installed or running? What if PopClip is installed but not running? or running but with “Appear automatically” turned off?
Let’s see if we can dig into it a bit with more information. When CPU usage goes through the roof, which application is showing excessive CPU usage in Activity Monitor? Is PopClip, the System Settings application itself, or something else?
Regarding the missing accessibility permission, I guess this could be related. Ane thing I can think to try is maybe manually add PopClip to the accessibility list using the + button,
One other idea: try adding System Settings to PopClip’s excluded app list. Does it make any difference?
You mention Klondike forever (game?) slowing down. I wonder if this game could actually be indirectly the cause of the issue. What happens if you quit that game?