Pitch Attitude vs. Pitching Moment
Care must be taken to distinguish between pitch attitude and pitching moment. Changing the CP (fore / aft) will induce a pitching moment change that must be balanced via the tail (elevator and/or stabilizer). The title of this thread, however, makes reference to pitch attitude, not pitching moment.
Trim pitch attitude for an airplane is more a function of the lift curve than the pitching moment curve. Extending flaps changes the Cl vs. Alpha characteristics by increasing the lift generated at a given speed / AOA. Whether flap extension moves the CP forward or aft, it will increase lift at a given AOA (assuming speed is held constant). As a result, trim pitch attitude will be lower following flap extension regardless of what happens to the CP. Note that on some airplanes this effect is accounted for within the control system such that flap position changes are directly compensated via commanded pitch attitude change to maintain lift and thus avoid disturbing the flight path angle.