Flight Instructor reported loss of power when attempting to make a simulated engine out landing during a touch and go. The Instructor lowered the landing gear and the aircraft touched back down in the grass past the runway.

Narrative

I was conducting a check ride review flight for the commercial multi-engine rating with my student. We were doing a cross country scenario from ZZZ1 to ZZZ2. Just south of ZZZ; I failed the right engine for a training exercise. My student executed the memory item. he then ran the engine securing checklist and fully shut down the right engine. He began diverting to ZZZ. Shortly there after; I instructed him to run the unfeathering procedure. After; I placed the right engine in simulated feather; which consisted of 11 inches of manifold and 2;000 RPM. I told my student to only use the left throttle to land. We entered the pattern and put the gear down abeam the landing point for Runway XX. We turned base and final. On final; I moved the right prop to full in the event we needed to do a go around. We landed; and as planned; proceeded for a touch and go. My student retracted the flaps and moved both throttles to full. Upon doing so; the aircraft moved right on center line. I took controls; rotated; and raised the gear. I realized we were unable to climb or increase airspeed. I then put the gear back down and pulled both throttles to idle and landed straight ahead. We touched down in the grass just off the end of Runway XX. After we came to a stop; I asked my student if he was ok. He said yes. Then I told him to shut it down. He turned off all switches and we exited the aircraft. I called ZZZ1 to tell them what happened. My Chief Flight Instructor arrived; took pictures; and called rescue services.