GA Flight Instructor reported a near miss with another aircraft in the traffic pattern at GAI non-towered airport on a training flight. Reportedly; the other aircraft did not fly a standard traffic pattern and created the traffic conflict.

Narrative

I was the MEI in Aircraft X on a training flight; conducting an RNAV A circling approach at a busy; uncontrolled training airport. We extended our downwind leg to allow spacing behind a jet on final. While turning final; a small aircraft pilot--who had reportedly 'had us in sight'--called and asked to go missed. I misunderstood; assuming he was on the ground referring to a void time; and replied: 'Negative; short final; simulating single engine landing.'Shortly after; the small aircraft cut inside our base; flying above us without establishing proper radio communication or visual coordination. the runway was still occupied; so we initiated a go-around. During the upwind; I identified the small aircraft directly above us--he had crossed without separation; failed to side-step; and violated standard traffic pattern procedures.This resulted in a near mid-air collision. The other aircraft did not apply proper see-and-avoid techniques or fly a predictable pattern. I'm submitting this report to emphasize the critical importance of flying standard patterns; maintaining clear communication; and respecting traffic on final--especially in a high-density VFR environment.