C172 Flight Instructor reported losing engine power on takeoff and needing to land as soon as possible; informed ATC that the aircraft will overshoot the runway and roll into the grass.

Narrative

Acting as a Flight Instructor; conducting an instrument lesson with a student on a VMC day. The plane was in the hangar overnight. Everything during preflight; and start-up through run-up was normal with the aircraft. We inspected fuel visually in the fuel tanks using a fuel stick and sumped. Magneto check; RPM dropped 100 on left and right. Carburetor check; RPM dropped with no rise. All gauges within limits and followed checklist. On takeoff roll on Runway XX; gauges were within limits and airspeed came alive. We rotated at 55 and began climbing out at Vy. Everything sounded normal; all gauges looked green; we had a positive rate climb. A little over halfway down the runway around 200 ft. AGL - rough estimate; I'm not 100% sure on our altitude before aborting takeoff - the engine began shuddering; and I looked to my student; who looked to me; confirming we both felt and heard the engine roughness. I took control and looked in front of us and inspected the length of the remaining runway. I thought to myself; 'If I abort now; I may overshoot the end of the runway into the grass; but if I continue and try to troubleshoot and the engine gets worse or fails; then we are headed for trees.' I also think quickly that the cause could be carburetor icing; but if I pull carb heat and continue with the climb and the engine does not improve; then I have set myself up for a bad scenario. At this point I think it is better safe than sorry and pull power to idle; begin a forward slip and add full flaps; hoping to get down to the pavement in time. I announced to ZZZ Tower; 'Aircraft X aborting takeoff loss of RPM;' although; I unfortunately did not look at the engine gauges as my biggest concern was landing the aircraft safely at that time. Tower cancelled the takeoff clearance for the aircraft behind us. As we were descending I realized we would indeed overshoot the end of the runway so I announced to Tower that we would overshoot the runway and roll into the grass; Tower acknowledged this. We touched down about 5 - 10 ft. before the end of the pavement and rolled into the grass. I kept the yoke pulled back to hold off the nose wheel as much as possible. The grass was pretty tall and helped dampen the roll and bring us to a smooth stop. Once at a stop I confirmed my student was okay; took a brief look at the gauges; which looked green - although I wasn't looking very thoroughly here; again we had just stopped - and then announced to Tower that we were at a stop in the grass and us and the plane were okay. Tower asked to confirm and I reconfirmed and told Tower I would be shutting down the plane now; and Tower responded okay and that service vehicles were on the way. We shut down the plane as normal. The ground check on the ignition was normal before shutdown. We put the cowl plugs on; the pitot tube cover on; and the yoke control lock in. I called Dispatch and informed them on what happened. An airport vehicle arrived and they confirmed we were alright.