PA-28 Flight Instructor reported getting close to an IFR aircraft while conducting maneuvers for an engine break-in training flight at night.

Narrative

The flight being conducted on this night was an engine break-in training flight. These flights require us to remain below a density altitude of 5000 ft. and keep the aircraft power setting high in order to properly seal engine components. Due to the criteria; we do not always request flight following to provide support as we cannot always comply with the assigned altitudes changes. This night we were navigating from ZZZ2 to the ZZZ3 area in VFR conditions. It was night time just north and over populated areas. There were ground lights; lighted mountains; and lights from aircraft; but otherwise no potential visibility restrictions/conflicts due to weather. We navigated north of ZZZ4; to ZZZ5; north of ZZZ6; over the top ZZZ; and south of ZZZ1. The flight did continue eastbound just short of ZZZ3 with a westbound course reversal. However; the event occurred near the ZZZ / ZZZ1 airports during the initial passing. To have best situational awareness; we monitored each airport Tower or CTAF frequency prior to and passing the field. We were heading eastbound near ZZZ and saw two landing lights from aircraft maybe 5 - 10 miles in front of us; near the ZZZ1 Airport. I believe one was transmitting on the ADS-B and one was not; but to be cautious; we had repositioned our aircraft to pass in between the two lights. This altered our intended course north of ZZZ to be overflying midfield. When the traffic in front of us had been noted; we had swapped to monitor ZZZ UNICOM in an attempt to listen for them and their intentions. After a minute or so of discussing the potential aircraft in front of us; the copilot in the right seat started pitching down into a descent. He noted that traffic was passing directly above us and I could only see them if I sat all the way forward in my seat looking directly up. The aircraft was not on radar and the traffic avoidance system onboard our aircraft never chimed to indicate a close target. The system typically reaches a few miles out; indicates a clock position; and includes if the traffic is high or low. The aircraft that passed over the top appeared to be on a north-northeast heading and seemed to pass from behind; with an altitude 100 to 200 ft. above us.We were following the easterly cruising altitude of 3;500 ft. at the time with small fluctuations for mild chop; descending down between 3;400 ft. to 3;350 ft. once the right seat pilot spotted the traffic above. The copilot looked into the flight after; noting it was an instrument departure out of ZZZ. I do not recall the tail number of the aircraft. It is unknown if they were already in communication with ZZZ Departure or if they had only received a clearance void time; departed; and were waiting to establish contact. With this information I believe the aircraft was slightly behind us and ultimately ascended above our altitude to execute the departure procedure. I am unsure if they had ADS-B In; but the transponder was on for the entirety of our flight.