Flight instructor with student reported taking evasive action to avoid a near midair collision in the traffic pattern at a non-towered airport.

Narrative

Had just departed runway XX @ ZZZ. 3-4 other airplanes in the pattern properly making traffic calls. After a touch and go landing we made an 'upwind of runway XX' call. I noticed on our ADS-B-IN system; an aircraft with a head on trajectory about 3 miles South; but due to the haze; did not have the aircraft in sight. I advised my student to turn crosswind and made the call to CTAF 'on crosswind for runway XX'. There were two other aircraft doing simulated instrument approaches. Piper Archer decided to practice a RNAV XY; circle to land. I did not hear a circling or joining downwind call from the Archer. My ADSB indicated this was the aircraft inbound circling; and I was vigilantly visually searching for this aircraft as we were climbing. At about 500 feet laterally; I saw we were on an imminent collision path with the Archer. I took the controls from my student and pitched up significantly to avoid collision. We passed above at about 200 feet. I then asked on CTAF why they did not make a call about joining the downwind on a straight in and expressed how dangerous the situation was. Their response was that they had made an approach call some 8 miles out and 'were on a checkride and had to circle within 1.3 miles of the airport'. We promptly left the area.