Flight Instructor reported they had a near mid-air collision at a non-towered airport with an aircraft that was on final approach to hook a banner.

Narrative

Initially; during a descent to land; Aircraft Y departed from runway intersection 1 onto runway XX for departure. Our aircraft was conducting a power off 180 and was around 300-400 feet away from the end of the runway when Aircraft Y pulled out in front of us. I made a radio call declaring that we were short final for runway XX; and was ignored and watched Aircraft Y depart the runway in front of us; we executed a go-around procedure and climbed out to execute our crosswind turn into the traffic pattern. He made an initial traffic pass and missed his banner hook the first attempt; and swung around behind us in the pattern after we had completed another landing. We executed a landing and then began our takeoff like we declared for our touch and go; and receive a TA warning from the TCAS unit detailing that we have traffic at our altitude less than 100 feet away from us horizontally; we climbed out below Aircraft Y glidepath as to avoid climbing into him; but intentionally started a left hand crosswind turn before reaching 500' AGL to avoid potentially colliding with the banner he picked up. He then departed the pattern after the LOS. During the incident and his time during the pattern; Aircraft Y made a few radio calls detailing his position in the pattern; but frequently 'stepped on' others that were already speaking on frequency; or made radio calls that were difficult to hear on frequency. We did not receive a radio call about his intentions to pick up his banner until we had already rotated during out takeoff; leaving no time for decision making about whether or not to conduct a full stop/stop and go takeoff.