Upon further research, I've managed to find more definitive evidence as well as many more pictures of such weapons. A Turkish auction house sold some of them, both caplocks and flintlocks, recently where they were described as "Zeytun Ajars", with this being their reported contemporary name:
https://www.egemuzayede.com/urun/616...zeytun-acari-o
So all this evidence combined seems to suggest quite definitively that these weapons are of a Zeytun Armenian origin. As far as uniquely Armenian style weapons go, these Zeytun Ajars are one of only two to three that I know of (With the others being Armenian style Kindjals and possibly certain Yatagans), so they hold quite a special place.
More pictures down below (All credit to Auctions Imperial and Eg Muzayede) as well as a picture of the Princes of Zeytun wielding these weapons.