What company's staff checks boarding passes at the gates at Tbilisi airport?

I'm soon flying from Tbilisi (TBS) to London (LHR) via Doha (DOH) on Qatar Airways (because of a fare glitch - it's a single booking). As my passport is getting replaced, I will only have access to my ID card, with which I can exit Georgia and enter the UK, but not enter Qatar (which I don't intend to either)

Since Timatic says nothing about whether I need a passport simply to change planes at DOH, I called the Qatar Airways UK hotline, who told me I was fine to travel. I also gave the IATA travel centre a shot - it said I was good to go.

Unfortunately, Qatar Airways refused to send me a confirmation via e-mail, instead saying I should print the Timatic extract to show staff at TBS if necessary.

The problem is that Timatic says nothing about this specific issue, and in my own experience, most gate staff are reluctant to let you fly somewhere even for transit without a document accepted for entering that country.

So, what I need to know is who checks boarding passes at the gate at TBS, because I need to contact them to clear this up immediately