SunExpress - 3+ hour delay because of their systems being down, but SunExpress are claiming it wasn't their fault

We were booked on SunExpress XQ584 from Antalya to Bristol on 15 August 2026. I was actually meant to be on the XQ584 the prior day but I spent 45 minutes queueing up to checkin to be told the check in is already closed...but that's mainly my fault!

Anyway, while waiting to be board we noticed a lack of any SunExpress staff near the gate. Time went on and our departure time came and went, and in the end we heard that SunExpress's systems are down. The gate next to us was for other airlines and they were boarding and taking over as normal. Eventutally after over 3 hours, we boarded and had a bit of a wait to take off due to SunExpress having to coordinate certain things. The flight was scheduled to arrive in Bristol at 09:35 but actually arrived at 12:41, so the arrival delay was 186 minutes (3 hours 6 minutes). I heard some people on the plane mentioning they were glad it was over 3 hours, and I later discovered the compensation policy where you are entitled to €400 per passenger if the plane is over 3 hours late to arrive...

I logged a request to SunExpress regarding this, and they give me a vague response about it possibly not being there fault and advised me to contact the 'Directorate General Of Civil Aviation' in Turkey, which I did. I've now heard back from SHGM who have rejected my claim. It states that SunExpress has said:

"Our review showed that the delay was caused by an external technical disruption affecting operational systems, which was beyond the airline's control. As a result, flight preparation and passenger handling processes were disrupted."

SHGM has now closed the complaint and accepted SunExpress's explanation, saying the event was force majeure and therefore no compensation is payable. I have no idea how they have come to that conclusion, given that there is evidence that other airlines were operating fine at this time - it was just SunExpress. In fact, on the day SunExpress published an operational announcement that day saying: "Due to a technical issue affecting our systems earlier today, Check-in and boarding processes were temporarily disrupted."

I've gone back to SunExpress asking for full details on what the disruption was, and asked for evidence regarding this. But I don't expect to get anywhere to be honest, but I also don't really want to give up. Does anyone have any advice on how to proceed, or am I chasing something I likely won't get?