GA flight instructor reported making a go-around to avoid departing traffic from intersecting runway resulted in an NMAC.

Narrative

We (Aircraft X) were short final (about 400ft AGL) for Runway 10 at Sebastian Municipal Airport X26 when Aircraft Y made a call saying he was departing Runway 05; which intersects Runway 10. Until then; my student had been making all the required pattern leg callouts; along with Aircraft Z that was also in the pattern with us; also making callouts. I made a call back questioning that he was departing Runway 05 at Sebastian (X26) as a few other un-towered airports share the CTAF frequency 123.05 and also informed him we were short final for 10. Aircraft Y responded; but continued his departure. It was at this time that I told my student to go around; we were 200 ft AGL at this point. We started our climb as I kept visual separation between us and Aircraft Y. Aircraft Y rotated and started their climb out. The closest we came to them was at the intersection of Runway 10 and 05; passing over top of them at 400 AGL while Aircraft Y was about 100-200 ft AGL. We kept climbing on the upwind Runway 10 while Aircraft Y turned right heading 100 to fly parallel to us while still climbing; still about 300 ft below us. We ended up having to climb to 1500 ft AGL before turning left crosswind to avoid Aircraft Y who had leveled off at 1000 ft AGL. Aircraft Y then made a right crosswind departure to fly southbound down the shoreline which is non standard as Runway 05 is left traffic.A pilot in a canopy top airplane on the ground was witness to this whole event and has submitted his own report of these events. He also has pictures of the ADS-B from the climb out after the near miss if needed.