C172 instructor reported a NMAC during the circling to land maneuver after an RNAV approach. Instructor executed an evasive maneuver to avoid the other aircraft who departed the traffic pattern in a non standard manner.

Narrative

My student and I were flying an instrument flight lesson. We started by going up to ZZZ shooting RNAV XX to the missed. Then was going to to VOR-A into ZZZ1 but ZZZ [VOR] was down so we decided to fly the RNAV XY in ZZZ1 circle to land Runway XZ. Being a clear day there were quite a few aircraft out so we were making proper radio calls on the way down. Winds calm at ZZZ1 people were taking off Runway XZ. The piper aircraft made a radio call for takeoff and the instructor pilot had him in sight throughout this whole incident. The piper climbed to traffic pattern altitude and waited until roughly three miles until turning crosswind so I assumed he was departing to the north. I told my student to start his circle to land early for safety reasons. We started the circling and he called turning crosswind. At this time as the instructor; I took control and kept us at 2300 feet MSL to maintain well above the traffic pattern. He then started to climb in our direction as he was planning on departing off of the downwind. At this moment I maneuvered us clear and then proceeded to let the student come in and land the aircraft.