Flight Instructor with student reported the engine began running rough during initial climb after a touch-and-go landing. Flight Instructor made an immediate turnback to the airport and a precautionary landing on a taxiway.

Narrative

My student and I flew to ZZZ1 where we executed multiple successful touch and goes. We experienced very slight engine roughness during one of our takeoffs from ZZZ1 for a brief second due to my student advancing the throttle too quickly. Other than that; nothing was out of the ordinary. After a few landings we departed for one more touch and go at ZZZ.The approach into ZZZ was normal; and we had no indications of engine trouble. After touching down on Runway X; my student added full power to takeoff; forgetting to remove the flaps beforehand. Flaps were set 20 degrees; so our initial airspeed after lift off was very low (somewhere around 50 KIAS). This was the first moment I came on the controls; only to keep my student from pitching up anymore; and to keep him from removing the flaps before we were at a safe airspeed/altitude to do so. Still assisting my student with slight forward pressure on the yoke to allow airspeed to increase; I removed 10 degrees of flaps. As soon as I set the flaps from 20 to 10 degrees; the engine began running extremely rough and we experienced a partial loss of engine power. I cannot recall the exact altitude; but we were roughly 400-500 ft. AGL. So that leaves us at a very low altitude; with little to no runway left ahead of us; and a decrease in lift from reducing flaps just before the loss of power occurred.I immediately took over the flight controls. I noticed we went from a respectable climb rate to only maintaining level (at full power). I verified the fuel selector was on both; fuel mixture was full rich; and checked engine instruments. Oil pressure read slightly high; nothing catastrophic. However; the roughness sounded and felt to be getting worse; and I was not confident that we would be able to maintain altitude much longer. Due to our critically low altitude; and the uncertainty of whether or not the situation would deteriorate further in the next few moments; I made the decision to execute a precautionary landing; while the possibility of landing on a runway still existed. I informed ZZZ Tower that we were experiencing engine roughness and stated that I was turning around to land Runway XY. At the time I made that call; I did not declare an emergency because it was only engine roughness; and although it was the worst engine roughness I've ever heard; and I was fearful that the engine would give out - it hadn't yet and we were still making power. I was prepared to declare an emergency if the situation got any worse; but after the point when I began my turn there was no time to do anything except fly the airplane.I made a left turn into the wind to begin my descent for Runway XY. Losing altitude in the turn; and considering load factor/airspeed; I turned with the minimum radius possible while avoiding a stall/spin situation. The stall horn came on for a moment; and it became clear that we were not going to make the runway. I had a few moments left to consider the remaining options: land on Taxiway 1; or try to clear the GA ramp/Terminal to land in the grass beyond them. The Taxiway was the safest option; considering no other aircraft were occupying it at the time; and Runway XZ-XA is closed. I touched down on Taxiway 1; brought the airplane to a stop; and regathered my thoughts. Tower asked if our engine was still operational; to which I answered yes. I offered apologies to the ZZZ Tower for landing on Taxiway 1; because as mentioned above; I did not have enough time to declare an emergency/update them on my situation once I began the descent to land. I requested to taxi to the GA ramp and shutdown.Final words:If I could have a second chance on dealing with this scenario; I would have done everything exactly the same. As much as I wanted to fly a pattern for Runway X and do a precautionary landing with a headwind; that option did not seem to be on the table. My choices were:Enter a downwind for Runway X and hope we keep the limited performance we have to make it all the way around the pattern; orLand the plane as soon as possible before it turns into a complete power lossI chose the latter and would do it again.