SR20 Flight Instructor reported rough running engine resulted in priority handling return to airport.

Narrative

At the time of the event; instructor and student were practicing commercial maneuvers including lazy 8s; stalls; and slow flight. We were in the northwest practice area talking to ZZZ Approach at either 4500 MSL or 4000 MSL. After student demonstrated Power On Stall in a turn to the full break; and completing recovery ( PUSH; ROLL; THRUST; STABILIZED); instructor took the flight controls because instructor want to re- demonstrate maneuver since student didn't keep bank that was specified by the instructor (instructor requested 10 degrees and student went to 20). It was at this point instructor and student both noticed the yoke starting to vibrate and slowly begin shaking. After confirming with each other what was being noticed; we then realized the engine was sputtering too. Instructor and student confirmed the fuel pump was on; and then decided to enrichen mixture in case it was too leaned out (mixture was already at the sienin/blue line at the time). We also switched fuel tanks too. ZZZ Approach then came on and asked us to go somewhere else but instructor said 'unable at this time; we're having some engine issues.' Since we were high enough in altitude and already heading towards the east towards civilianization; Instructor had student pull out checklist for engine failure and began running memory items while student confirmed it with checklist. Engine was continuing to sputter and at this point was at full power; pitching slightly up (between 2 and 5 degrees); and indicting 118 kts. Instructor tried decreasing power slightly to 70% to see if sputtering would stop but it didn't change anything other than cause plane to decrease in altitude (100 ft.). Instructor added full power again and told ZZZ 'we need to [request priority handling]; we're having engine problems and we don't know if the engine could die on us'. ZZZ asked do you want to go back to ZZZ2?'; Instructor said 'negative we would like to go to ZZZ1; but where is the nearest airport'. ZZZ was the option given to us so student and instructor requested to go there and land there. Instructor then had student confirm with checklist again to be sure we definitely did everything; review any other checklist applicable to situation; and began running decent checklist and landing checklist. Instructor also told student to have open and ready CAPs checklist if engine were to fail. Instructor also had student get the comms ready for ZZZ Tower. Once cleared to ZZZ airspace and in contact with ZZZ Tower; ZZZ Tower said we're cleared to land on XR and that they will divert the other aircraft away from us. ZZZ Tower also asked us if we would prefer to land at ZZZ1. We were approximately at 3500 MSL at this point and within 2 miles of the runways. After discussing with students the option; the engine sputtering had final deceased. It was sputtering for more than 5 mins at this point but less than 10 minutes (approximately 8 minutes if I had to guess). Since engine sputtering had stop and started to operate normally; student and instructor decided since we had altitude to possibly turn back around if needed or continue to land at ZZZ1 ( or even worse pull CAPS) that we would continue flying to ZZZ1 and hold current altitude at 3500 MSL. We were then handed off to ZZZ1 Tower whom then cleared the runway for us to land. Instructor once seeing we had enough altitude to land finally reduced power and begin descend from 3500 for the straight in land on Runway XX. Instructor added fuel flaps and began slipping down to runway; only adding power once for a second to see if we still had power which we did. With all items completed on landing checklist; we landed safely on Runway XX and were able to taxi off and head to parking. I don't know what else we should have done; I felt both my student and myself operated at the highest standards of safety given our situation. Please let us know if there is anything more or less we should have done. I could not be more proud of student who remained calm during the whole situation and did everything instructor asked of them. Student aided with decision making process and we acted as a crew."