I was having similar issues, and haven't managed to trace the root cause yet. Mine would randomly start trimming down for no apparent reason.
@Jake@BevanAviation was very, very, very helpful in tracking down the gremlins, but weather got in the way.
I would suggest you start with a through and proper ground test (as suggested to me by Jake):
With a level attitude gyro engage FD and A/P on.
Hold CWS and position the yoke to about mid travel then release CWS.
While supporting the yoke use the up or down modifier button in the KC-290 to provide a pitch command.
Resist the movement of the servo and see how long it takes for auto trim to start.
Note the effort on the yoke from the pitch servo and the direction of auto-trim.
Hit CWS and reposition the yoke if needed.
Do the same test but in the opposite direction and note the results.
Hit CWS and reposition the yoke.
Without using the Up/Down modifier in the KC-290 apply pressure on the yoke with a push/pull for constant pressure.
Note the time it take for auto trim and the direction.
Repeat the test in the opposite direction and note the results.
Quoting Jake: When supporting the yoke with no input command from the KC-290 or pressure on the yoke there should be no auto-trim. If you notice a disparity in the effort needed to start auto trim we might have a switch or spring that has moved inside the pitch servo. For this installation the pitch servo needs 11inlb of torque to close a micro switch per the MM for the servo.
Also, make sure you're doing this on level ground, so that the plane's attitude is "sane" (no bank). Pitch will, obviously, be different than in cruise.
Interestingly, my AP, also a KFC200, also in a M20K, never beeped during the self tests, but I didn't trace that down yet, either.
For me, I'm focusing on some stiffness in the trim jack screw - it takes more force to turn the manual trim in some spots / trim settings than in others, and I'm thinking this might be confusing the auto-trim feature.
Good luck!