My personal opinions only...
I think a German LCC tried a route from western Germany to Tbilisi for a few years last decade. The real problems are:
1 - From the Caucasus to western Europe (i.e. where the big money is) is a long way, rather further than the average optimal distance for a profitable LCC flight. That means there needs to be a particularly good reason to fly to these destinations. Luxor as an example, has the attractions of ancient Egypt, the Red Sea offers beaches and diving on the cheap.
2 - A few years ago, I spent 2 weeks as a tourist on my own going round the Caucasus. There's not actually *that* much there to see, and promotion of the area as a tourist destination in the UK at least is poor
3 - It's not part of the EU, so traffic rights are rather limited and the standard join-the-dots route to profitability is closed off. Easyjet could fly there only from the UK and Switzerland. Ryanair can fly only from Ireland - too small a market to be worth trying. Air Berlin or Germanwings might want to give a Tbilisi - Berlin / Dusseldorf / Cologne route a go, but that's about it. Perhaps one of the Caucasus countries want to let Wizz set up a subsidiary airline similiar to the Ukraine, but would Wizz have confidence that there would be no back-room nobbling going on by a local airline ?
4 - Obvious as it is, until the Armenia - Turkey and Armenia - Azerbaijan borders reopen, or Iran becomes a friend of western Europe, Armenia will remain a backwater.
I think the Caucasus will eventually come onto the LCC map, but it'll take quite a while. Places like Tunisia, central / eastern Turkey, and the Ukraine will possibly come further up the priority list.