A Hero of Our Time / Герой нашего времени, by Mikhail Lermontov
Scoring nearly as well on LibraryThing, though doing a bit worse on GR, is a famous play which I really should have thought of myself. It was written in German in 1944, but first performed in English translation in 1948. Almost all the characters have obviously Georgian names; what's more, we are told that they live in a country called "Grusinia" which is pretty obviously a variant name for Georgia. There is a minor problem in that the framing narrative of the play is set in a Caucasian village devastated by the Nazis, who never actually reached Georgia; but that framing narrative is not extensive (and is often dispensed with in performance). There is a major problem in that most of the action takes place in the city of Nukha, which was indeed part of various Georgian states during the medieval period, but is now called Şəki or Shaki and is firmly in the territory of today's Azerbaijan. So with even more regret, I fear I must disqualify:
The Caucasian Chalk Circle / Der Kaukasische Kreidekreis, by Bertolt Brecht
That does leave me with a clear winner out of the three, one of a series of urban fantasy novels set in a post-apocalyptic future where magic has returned to the world, and not in a good way. The series as a whole is (I am told) set in the wrong Georgia, specifically in Atlanta. But in the sixth volume, our heroine and her allies venture east to Gagra, which is a real place in Abkhazia, and spend most of the book there. I'm aware of the views of local residents, but the fact is that Abkhazia is still regarded as part of Georgia by most international actors. So, with apologies to David Turashvili and Wendell Steavenson (and indeed to all local residents both sides of the Inguri river), as far as I can tell the best-known book set in Georgia or Abkhazia is:
Magic Rises, book 6 in the Kate Daniels series by "Ilona Andrews" [Ilona and Andrew Gordon]
Thanks to those who contributed to the discussions.