That one has a “Century IIB autopilot turn and bank” which is interesting in being a rather basic AP but with a yaw damper!
I have the KAP150 on the Mooney with yaw damper, I have two modes on the yaw damper ‘ON’ or ‘AUTO’ (and OFF…). I usualy select ‘ON’ and then add some rudder input throught the yaw damper with a knob when/if required, but I never used any other system.
it may be that the plane is out of trim in the yaw axis. There is a little trim tab attached to the rudder. If you have to input right rudder to center the ball, you need to bend the tab to the left. This causes a deflection of the rudder to the right, which is what you need.
DA42 with GFC700 may have yaw damper
Yes, I have flown a DA42 with GFC700 which had a yaw damper.
never saw a DA40 G1000 with a KAP150
neither did I, I think it is a KAP140. I used to fly a DA40 ten years ago with this AP, but it had no yaw damper.
FWIW we have a KFC150 in our Mooney with a yaw damper but it’s an Stec. It’s a completely independent unit.
Are you sure it’s a KAP 150?? (never saw a DA40 G1000 with a KAP150) morelikly a KAP 140 ??
As far as I know the KAP 140 it cannot have a yaw damper.
manualy adding rudder seems the best way or let the maintenace organisation adjust the rudder trim tab,
DA42 with GFC700 may have yaw damper. DA62 has it.
the DA42 and at least some SR22s have it
Probably not with a KAP150 
The YD kit for the King autopilots is described e.g. in the KAP150 Pilot’s Guide and it is a number of pricey bits
A rudder trim is a different thing, and for sure you can use a rudder trim while a normal autopilot (no yaw damper) is engaged.
On their website I find ‘optional on some aircraft’.
Since the DA40 is not the ultimate top-end aircraft let’s assume it is not installed.
The DA40 also does not have rudder trim.
I should check if there is a trim tab I can adjust at the rudder..
The one I fly doesn’t. Leg sometimes starts shaking when reaching FL120
Peter: the DA42 and at least some SR22s have it
