I'm not sure why you're having difficulty finding "any information" about bike rental; I found several companies in each of Georgia and Armenia, although I didn't go to Azerbaijan. I ended up with Mototravel Georgia and Dreamriders Armenia, and had good experiences with each. Notable: Armenian rentals were much cheaper. I have some doubts you'll find a 1200GS, but maybe. Definitely make your arrangements, including deposits on specific bikes for specific dates, long in advance.
I did not find any way to take rented bikes across borders, which does make perfect sense. There are guided motorbike tours which take in Georgia and Armenia (and perhaps Azerbaijan), if that's something you'd be interested in.
Great countries for riding, with main roads in pretty good condition, secondary roads generally quite fun, and remote roads thoroughly challenging. Lots of history (including some very recent), lots of spectacular scenery. Main safety concern: large, aggressive, fiercely protective dogs in the mountains. Do not act threateningly toward any sheep!
Hope that's helpful.
Mark
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