C-150 flight Instructor reported an NMAC event during landing pattern training with another training aircraft. Both aircraft lost separation during the crosswind turn resulting in the C-150 Instructor electing to depart the pattern.

Narrative

My Student and I were holding short of the departing runway (Runway XX) at the ZZZ. There were three aircraft on final for Runway XX - a Tomahawk; C172; and PA-44; respectively. We had all three aircraft in sight and displayed on our TCAS. As we waited for the first two airplanes to land; we noted the time and distance it took for each aircraft to cross the runway threshold. Based on this information; we estimated the third aircraft's (PA-44) distance from the runway would allow us make a normal takeoff without causing any hazard to us or the approaching aircraft. To confirm our assumption; we asked the PA-44 for their approach speed. They responded; '100 kts.' Based on their speed; distance displayed on TCAS; and distance perceived to our eyes; my Student and I both agreed that we had enough space to make a safe takeoff. As the C172 touched down; I announced on the common traffic advisory frequency (CTAF) that we would 'line up and wait on Runway XX for the C172 to clear the runway.' We planned to start our takeoff roll as soon as the C172 exited the runway. As we taxied onto the runway; we noted the PA-44's distance still sufficient to make a safe takeoff. The time to cross the runway hold short line and the runway's centerline took less than 10 seconds. The C172 exited the runway about the same time that we lined up on the runway. Without delay; my Student commenced our takeoff roll. Upon reaching our rotation speed (65 kts.); we heard the PA-44 announce their go-around. We monitored the PA-44's location while we climbed upwind. As we climbed and crossed the runway's departure threshold; I noted the PA-44 slightly to the right and crossing the runway's approach threshold - a distance of about 5;264 ft.; or just under one mile between us. As we climbed upwind; the distance between our position and the PA-44 was decreasing. When we turned crosswind to remain in the pattern; the PA-44 also turned crosswind; placing them directly behind our position. As we continued to climb to pattern altitude (1;800 ft.); our TCAS display showed the PA-44 already at pattern altitude; behind us; and closing in on our position. I keyed the microphone and requested the PA-44's intention to remain or exit the pattern. The PA-44 did not respond. When the TCAS display showed the PA-44 almost directly above us and at a difference of 300 ft. in altitude; my Student and I elected to level off and exit the pattern at/around 1400 ft. to the southeast. We transmitted our intentions and left the pattern. No further incident took place.