Flight instructor on training flight with student reported autopilot malfunction affecting pitch trim. Instructor dis-engaged autopilot and regained aircraft control.

Narrative

I was acting as a flight instructor under FAR part 61 flying with a student in a Cessna 172P. The aircraft was equipped with a Garmin GFC 500 autopilot for certified GA aircraft; as well as enabled auto-pitch trim. When the incident occurred we were climbing southbound; over the river; out of ZZZ airspace at 1;400 ft. MSL for 2;000 ft. As we were climbing; Student Pilot was pilot flying; with autopilot engaged; but then passed the flight controls to me (the instructor) to put on his view limiting device. I as the instructor was actively monitoring the autopilot when this occurred. The autopilot was engaged at ~800 ft. MSL and at 1;400 ft. MSL; while I was monitoring the autopilot; the aircraft immediately pitched down to a -20 degree pitch attitude nose dive. When this occurred; Autopilot was still engaged; so I immediately disconnected autopilot and attempted to bring the aircraft back to a level pitch attitude. When I went to pull back on the yoke; I realized that the aircraft was extremely nose heavy. I then immediately thought to check the manual pitch trim; which I found to be set to full nose down. So I immediately rolled the manual trim wheel back to relieve control pressure and pull the plane out of the nose dive. After I determined that we regained positive control of the aircraft; we continued climbing out of ZZZ airspace and continued on the flight with autopilot disengaged; without incident. It is important to note that the autopilot was in heading mode for lateral guidance; and indicated airspeed mode; set at 84 KIAS for horizontal guidance. I can confirm that we were indeed climbing at 84 KIAS and low airspeed/stall was not a factor for this event. This issue will be addressed by having a certified repair station review the aircraft and determine the cause of this incident. Looking back; I don't think that I as the instructor made the safest decision by continuing on the flight as planned. While we did regain positive control of the aircraft and flew the rest of the flight without issue; there could have been a re-occurrence of the incident because while autopilot was not enabled over the rest of the flight; electric trim was still enabled; and if the electric trim system was mis-installed; there may have been another runway trim occurrence even with the autopilot not engaged. Moving forward; if I ever experience a flight control issue; I will land as soon as practicable; even if I regain positive control of the aircraft.