Helicopter Flight Instructor reported while on instrument approach to a non-towered airport another aircraft cut across their flight path resulting in the helicopter taking evasive action to avoid a collision with the other aircraft.

Narrative

While on the RNAV XX approach into ZZZ; my students and I heard a radio call of an aircraft descending out of 3500 ft to the west of the airfield. Shortly afterwards; a medium-sized fixed-wing aircraft; one that is used in paradrop operations; cut across our approach path heading east of the airport. The fixed wing aircraft was in a steep dive at this time. I told the student to stop their descent and slow back to 60 kias. The aircraft continued to the east and dropped to a very low altitude in a matter of seconds. The next radio call stated he was 'WEST' of the field at 400 agl. This call was 30-45 seconds later. We could clearly see he was EAST of the field at a very low altitude. We were continuing on the approach and being vigilant. The aircraft cut across the airfield; ZZZ; at a 45 degree angle about mid field. At this point; the aircraft did a return to target; meaning they climbed up; banked hard to the left; did a 180 turn; dropped back down very low; and headed back to the east. Shortly afterwards; I heard the call that they were climbing out to the north to 3;500. I have seen this aircraft parked at ZZZ on a regular basis. I believe it is used there for the jump school. Once my student had completed the approach and we were on the missed approach climb out; I called the aircraft on CTAF and told him he was very close to us on our inbound approach. The pilot stated he was VFR and had priority. He was 'smart' and unprofessional in his communication. I responded that I know VFR is a priority; I was making the required radio calls; and there was no need to cut across the approach path like he did. He made numerous comments; one of which was that 'they would go back and talk about it'