Just finished reading
Romanianflyer's Trip Report! Many kudos to the author for such vivid and detailed descriptions and photos. Will definitely look forward to his remaining reports from that trip. Based on what I've read so far, I'd want to incorporate a visit to Ani into my travels as well. I'd also like to get close to Mount Ararat, although I'm not sure how much of that area is accessible to casual hiking.
In the process of looking up Kars, I came across references to the recently completed
Baku-Tbilisi-Kars Railway. Passenger train traffic is scheduled to begin in Q3-2019. Hope the actual start will be close to that target. I'm guessing that some of that delay is related to the new train technology of adjustable wheel base. This railroad uses sections of wide (Russian) and standard width track. In the past, moving a train between different gauge tracks required changing the wheels. Passenger cars were actually lifted and suspended during the process. Adjustable wheel technology will make all that unnecessary.
Like Romanianflyer, I would also prefer to avoid bus travel if a rail option existed. The promised BTK passenger service will be a very welcome development. If that happens, I would not mind skipping Batumi, as I've visited there in the past.