I remember, it was a Turkish bayonet. This still allows for the Laz origin; as a matter of fact, they might be produced by the same person ( or workshop). One can hardly get closer to the Caucasus per se than Lazistan, it is almost Minghrelia.
I am, however, puzzed by the 3rd one: apparently, there is a specific name for it in Kazakhstan. It would be unlikely to have a specific name for a totally foreign sword. And Kazakhs are Turkic people.....
These Laz Bichaks are the damnedest thins I know; the moment you think you nailed them, some new piece of information pops up!
Aral Sea yataghan?