C172 flight instructor had an NMAC with a Gyrocopter during approach for landing.

Narrative

At least 20 miles southwest of ZZZ; I checked the airport information on Foreflight and checked for NOTAMs and found none. I made my first radio call on UNICOM; 8 miles southwest of ZZZ with intention of doing a touch and go on the only Runway. I heard a gyrocopter call departing the pattern to the north. I made my next radio call 3 miles southwest; intending to fly northbound on the west side of the airport well outside the pattern; then make a 270 degree turn to enter the left traffic pattern on a 45 degree leg.I then heard another fixed wing aircraft report 6 miles west of the airport. I decided to abandon my initial plan and made a call to inform her that I would be entering the pattern on base leg to avoid any conflict with her. She (the fixed wing) acknowledged. I entered the pattern on a 1.5 mile base leg and then turned final and made a radio call on final. Approx 1/2 mile from the threshold; my partner pointed out a gyrocopter on left base cutting in front of us; about 200 feet above and to our left; turning a tight left base to final.It was a white; open cockpit gyrocopter. Our closest separation was approx 200 feet. I was at the lower altitude momentarily on final. I took evasive action by side-stepping right to the upwind leg to keep the gyro in sight as it landed. I performed a go-around on the upwind leg and exited the pattern straight out to the north. I heard radio traffic from the ground 'hey you just cut in front of that fixed wing' but no acknowledgment from the gyro pilot. We noticed that the airport grounds had a large number of trailers; maybe campers; and surmised they may be having a gyrocopter rally. I asked them on the radio if they had filed a NOTAM and they said yes; but the event doesn't start until tomorrow. On Date I checked the FAA NOTAM website and found none for the event. I called FSS and asked about NOTAMs for ZZZ and they had none. I asked how to file a near mid-air collision report. They said they could not take a report because gyrocopters are certificated aircraft and are not required to file a NOTAM for gatherings.They declined to take a near mid-air report. Seems to me they are making a judgment before even hearing the facts. I am filing this report because I think a NOTAM should be required for gatherings where a large number of NORDO aircraft might be present. I am also filing this report because the gyro pilots may try to find fault with me for not flying a full traffic pattern. I took the action I did (not flying a downwind) because 1) I heard no other aircraft on the radio in the pattern and 2) I wanted to avoid conflict and maintain spacing before the other fixed wing aircraft approaching the pattern - she acknowledged me and said I was in sight and she was number two.