So an update: Halifax agreed to process it as a chargeback. On DAY 45, Turkish Airlines decided to do a representment using a proforma reply. Halifax told me they send the same representment every time (it was factually incorrect, saying I had not requested a refund from their customer services, which I had, and which Halifax sent the email proof of).
Anyway, with another 2ish week delay before Halifax submit their rebuttal, it’s looking like October before this is resolved.
Absolutely shameless behaviour by Turkish Airlines.