I had no idea on HighFlyer's usage of FQTV and FQTS numbers until I chanced upon this thread in FT while trying to search for a solution.
However, there isn't a way for me check if FQTS was filled since BP and website will only have FQTV.

The ticket was reissued by customer service at some point after the initial booking and this may have wiped the FQTS - however, I did try to do my due diligence by emailing HighFlyer support after it was reissued and they had replied stating that they have checked and it was assigned for crediting to the company's HighFlyer account.

When it wasn't, I checked again and they gave multiple different excuses (as highlighted in my original post) and I'm not getting anywhere with Customer Service. Currently, all they're doing is using a partial read of 1.13 to cite that because I didn't credit to KrisFlyer, I'm not eligible for HighFlyer points and making egregious claims like "It is not possible for you to earn miles with Turkish Airlines and HighFlyer at the same time. You must credit your miles to KrisFlyer to get HighFlyer points"

The latest reply by the supervisor gaslighting is even more absurd stating that the FAQ item on HighFlyer's support page allows Corporate Travellers to credit their miles to other FFPs, but doesn't talk about HighFlyer points earning, hence, I'm wrong to interpret that HighFlyer points can be earned.