Flight Instructor with student pilot reported loss of control during final approach and landing resulted in runway excursion and damage to the aircraft and an airport sign.

Narrative

I was flying with a primary private pilot student coming in to land after a training flight. I was in the back and the student was in the front. This was the student's third training flight with me. The student had some prior aviation experience flying with his father. The student entered the down wind to land on runway XX at ZZZ. He was very close to the runway because he let the wind blow him closer to the runway. I told him to extend his downwind and crab into the wind to not get blown further into the runway. The student started to get slow; so when he turned base I took the aircraft from him and told him that I would fly the aircraft but that I wanted him to follow along on the controls. I added power and got the aircraft back to a good energy state and turned final. I ended up being a little high on final so slipped the aircraft a bit to get it back on 'glide slope'. Once on glide slope I set up to touch down just after the XX [numbers] on the runway. Just as I was preparing to round out a large gust of wind threw the aircraft up into the air. The student exclaimed 'whhhhoooohhh'!! Right after the exclamation I felt excessive control stick inputs; move the ailerons full deflection from one side to the other side. I said the the student 'I have the controls'. I got no response from the student but continue to feel control inputs; including back pressure on the stick. I yelled at the student 'when I say I have the controls - I have the controls!!!' Shortly after that that aircraft made a very abrupt turn to the left and departed the runway. I was concerned about the aircraft flipping over or standing up on the prop so I made gentle inputs to get the aircraft back onto the prepared surface. Unfortunately I forgot about the runway signs and the aircraft prop hit the side of a runway sign. The sign then hit the cowling. Once the aircraft got back onto the prepared surface there was a lot of vibrations so I taxied the aircraft off the runway onto a taxiway and asked the student to shut the aircraft down. An airport tenant drove me and the student back to the flight school. I called the flight school owner and she came out to the aircraft and took the necessary picture for the insurance company. While we were waiting for the flight school owner to show up I discussed what happened with the student; his friend a fellow student and the friend's mother. I asked the student why he got on the control. The student said he did not get on the controls. When the flight school owner arrived she had us both write statements for the insurance. In his statement the student said he did not touch the controls but saw the wings going up and down (left and right) right after the wind gust. After the student and his friend and mother left; the owner of the flight school and I went to look at the aircraft. We noticed that the trim level was pull all the way back. That would account for the back pressure on the stick that I felt. The next morning we drove out to the runway to see if we could see anything that would provide more information. We found a distinct line that went from the runway centerline and then made an abrupt turn to the left (just shy of 90 degree turn) to the left and then departed the runway. After seeing this and the full back trim my assumption is that the student got startled after the wind gust (as evidenced by the exclamation) and then started to flail about the cockpit first hitting the stick around from left to right and then trying to brace himself using the trim level which is on the left side of the front seat and pushing forward with his left foot. Pushing forward with his left foot made him depress the left brake (the aircraft has heel brakes). When he impacted the left back that caused the aircraft to dramatically shoot of to the left; leaving the black tailwheel mark on the runway. The flight school owner has submitted the claim to insurance and the claim has been processed. The school has also got an estimate to repair the aircraft and the school is proceeding with that work. The work will be covered by insurance.