score:8
Apparently TAV Georgia is the sole ground handling agent at Tbilisi airport.
Regarding the actual issue: I called Qatar Airways's central hotline and explained my situation, whereby they put a note in the system "IN TRANSIT TO LONDON HOLDING A NATIONAL ID CARD. OK TO BOARD".
Despite having checked in online and printed my boarding pass, I went to the check-in desk 4 hours before departure to get a written confirmation to show gate staff, as I wasn't sure they'd have access to the system like the check-in staff.
After 30 minutes of bickering with a lady about whether I could travel, and insisting that she summon her supervisor, she eventually did. He checked the system for 3 minutes, eventually asking me "you go to London?" I said yes, whereby he said "wait" and called someone. Then, he printed out two fresh boarding passes and put a special stamp on the Tbilisi-Doha one (presumably similar to Ryanair's and Wizz's stamps indicating a non-EU passenger has been cleared), and told me I could go to the gate.
When I presented the documents at the gate, the agent scrutinised the stamped boarding pass for 20 seconds, before saying "OK, have a good flight".
That was the end of it - the rest of the trip was 100% painless.