Flight Instructor reported a NMAC with another aircraft in the pattern at a non towered airport.

Narrative

While established on the left crosswind for Runway XX at ZZZ; another company aircraft made their initial inbound call at just 3 nm to the east of the airport. The student pilot; later discovered to be on a stage check; proclaimed he was planning for a midfield cross to the opposing runway (Runway XX). Immediately upon hearing the call; I (instructor pilot in the twin) made a radio call that the winds were heavily favoring Runway XY and we were already established in the pattern but did not have them in sight. The student in the approaching aircraft proceeded to make a radio call stating they had us in sight and were going to change to enter the 45 to the downwind we were at this point already on. The other aircraft then continued descend into us and made no turn to for their proposed entry. Not having them in sight; my student descended below pattern altitude where he caught glimpse of the incoming company plane directly at our 3 o'clock and made a turn to avoid conflict. During this time; we received at TCAS alert on the G1000 indicating just 300 ft. of vertical separation. Once appropriate spacing existed; we continued our pattern with no further problem.