Georgian Railway Map now generates a GTFS feed covering Georgian Railway routes and two international ones (Yerevan and Baku). It is updated at least once a week and is valid for 6 months ahead.
Soon I’ll try to add more information about trains that are not running at the moment. Georgian Railways only publishes it a few days in advance, but it looks like these trains can be reconstructed from previous years’ data.
The feed is available here: https://georailway.com/gtfs/georgian-railway.zip