We need to steer clear of tying AHRS to an AP out incident, if I remember correctly neither tarsyns or AHRs can influence the aircraft flight path if the AP is not engaged other than through the instrument-eye-hand-control route.
In SAS the ap will only be able to 'push' against the pilots stick position and therefore will always have a changing reference point, if there is control input being put in by the pilot the end effect on the flight path/attitude being unpredictable.
I believe that the mode of operation on this operation is not to use transparent piloting and either have force trim off or the button pressed. If so I seem to remember that pressing the button not only removes the 'fixed' reference point but also centres the AFCS actuators (check the API).
As ever awaiting the incoming!