A Light Sport Aircraft Instructor pilot reported loss of control due to a wind gust during landing resulted in a runway excursion and prop strike.

Narrative

Incident on runway XX at ZZZ. Student pilot was bringing us in for landing and began drifting too far left of centerline and losing airspeed. As the instructor; I notified the student that I was taking the flight controls and stabilized our approach around 4-500 feet AGL. Regained centerline and appropriate landing speed for runway 2 and entered ground effect. Once in ground effect; wind gust (10 gusting 15 from a 320 heading as memory serves); lifted nose of aircraft and resulted in drop in airspeed that forced airplane through ground effect. First touchdown of rear landing gear was hardly perceptible and I was not able to react quickly enough to initiate go-around to salvage landing attempt. Once porpoising began; wind gusts were causing left (wing side) wing to raise and blowing us off to the right side of the runway from left to right. Once I identified the porpoising; I attempted to maintain control of aircraft and centerline orientation; but couldn't arrest drift to the right side of the runway. Bounced 2 or 3 times while being blown off centerline; and felt it would be unsafe to add full power and attempt go-around with such a wing-high attitude for fear the additional power and sensitive flaperons could aggravate the situation and result in flipping the aircraft or further loss of control. I prioritized maintaining wings level while fighting for control of the aircraft; but could not regain centerline and our momentum/left quartering headwind carried us off the right side of the runway. Once the nosewheel entered the grass; it sunk in to the ground and stopped any forward motion; snapping the nose gear. I turned off all power sources and closed the fuel valve and myself and the student exited the aircraft awaiting the arrival of relevant personnel. No injuries to myself (PIC) or student pilot reported at that time; and no fluids leaking; presence of smoke or fire.