Flight Instructor reported on initial climb out experiencing a large loss of engine power resulting in return to the airport where they landed safely.

Narrative

While practicing the ZZZ Obstacle Departure Procedure (ODP) out of ZZZ from runway XXR; we had just taken our initial left turn to intercept the ZZZ [VOR] radial outbound; at 2800 msl and about 1300 AGL we had a large loss of engine power. At the time the student; a CFI training for CFII; was flying and I; instructor immediately took controls. I turned us into the left downwind for XXL the north runway. The initial power loss lasted about 5 seconds of the engine sputtering and rough running. The best way I can explain it is like the normal shutdown procedure when you bring the mixture to idle cut off. Again; after the first indication of an issue we had a positive 3 way change of flight controls as I turned downwind for the north runway. I communicated to the tower I was having an engine issue and making left traffic and would advise momentarily what I needed. By this point I had verified that the throttle and mixture were full forward; carb heat was off; fuel pump and mags all on; and that the fuel selector was fully seated in the right detent. The engine instruments were all in the green with the oil pressure around 60 psi; engine was making around 2000 rpm at this point. The engine seemed to come back to full power for about 5 seconds as I kept it at VY 76 to get as much altitude as possible. But after that we had another shutter and drop in rpms just prior to entering the normal downwind. The RPM was fluctuating rapidly; and it was like we would get steady rpms for 2 seconds then it would drop; come back; but then have a little less than the previous cycle. By this time; I had [requested priority handling] and the tower controller did a great job coordinating the 6 or so aircraft out of our way. Again; I had taken off XXR from the south side; and they offered me either XXL or XYR on the north side. With the throttle full and the power partial; at 76kts we only were losing about 100 fpm. With where I was in the pattern I felt comfortable making back around to runway XXL. By this point I had the CFI I was flying with reference the checklist and verify everything was correct. The only thing we did not do was swap fuel tanks. We had partial power loss and for the most part was able to maintain minimal altitude loss; so I felt swapping tanks came with the risk of making it worse. As I passed abeam the thousand foot markers; we were about 1000 agl so in the normal position for the power off 180 approaches we practice. By this time ATC had moved everyone off final and above us so I started the power off approach. I told the student I was with that I was starting the turn and if he thought I was getting high; low; fast; slow; or if anything started looking wrong; to communicate it to me. As we started the descent the engine got very quiet. We successfully and safely made the approach and touched down on the 1000 footers. The throttle was brought back and oddly enough the engine idled at 1000 rpms without issue. As I gave some throttle to get off the active; any setting over 1200 rpms resulted in rough running. We were able to limp it off of taxiway and onto the north ramp. We shut down and called our dispatch for a tow. The entire event was about 4-5 minutes from takeoff to touch down. Prior to take off we preformed the runup without issue; then held short for about 20 minutes waiting for takeoff. During that hold short time we did not swap tanks or do anything outside of the normal; before takeoff check list. On take off the engine instruments were all in the green like normal.