The descriptions of Georgia's devastation throughout the earlier parts of the chapter had me wondering about this specifically. The way the place is at the moment, there's a wide range of possible outcomes I could see the kingdom reaching under the suzerainty of the Komnenoi. Between the devastation of the population, the hollowing-out of the region's noble and princely houses, and the sudden power vacuum in the Caucasus and eastern Anatolia, David and his successors have a lot of leeway with regard to how they will integrate Georgia.For me, it's not how much he manages to conquer that will be critical, but way more how much he manages to integrate to the Trapezuntine realm and its system of administration.
Between the sheer desolation of Karvelia and the repeated mentions of Pontic soldiers having settled there even before David's invasion, I'm expecting at least a partial degree of Hellenization to occur in Georgia under the Komnenoi. How much ends up happening is probably highly dependent on how good times are in Pontus proper (periods of peace and stability producing overpopulation and a need to emigrate, civil war and/or plague causing the opposite), but if the kingdom ends up bound to the Trapezuntine state apparatus for a significant period of time I expect the majority of Greek settlement to target the lowland regions of Colchis given their abundant agricultural wealth and proximity to the coast.
At the same time, the foundation of any realistic Komnenian efforts to keep hold of Georgia will need to rely on ensuring loyalty among the Georgians rather than merely trying to colonize the region into submission. That might actually be a bit of a challenge in the medium-term, as with the Rumites reeling and the Qutlughids crashing there isn't an especially large threat from outside non-Orthodox polities. In the near term (barring extreme fumbles by David and/or his successor) the citizenry will probably be simply grateful by and large grateful to be relieved from anarchy and poor rulership, but these sorts of questions end up festering on the decadal and centennial scale. Then again, it's not unlikely that a reasonably strong and hostile power in northern Caucasia or Iran has nucleated by then and served as a binding force between Tbilisi and Trapeous.
Kartvelia could also well act as a place for younger sons of Trapezuntine noble families to gain new titles given the devastation of the Georgian ones, though simply raising minor provincial aristocrats to power or helping remaining native lines back to their perch (such as the Bagrations, something I personally hope happens in some form) would likely be just as if not more effective at making a Kartvelian noble class loyal to Trapezous. Then again, Roman politics being Roman politics, the long-term loyalty of provincial nobility is really more a hopeful suggestion rather than an achievable reality. The real question is dealing with the Eastern nobility, who seem to have gone through the war fairly unscathed while being highly heterogeneous and fractious between themselves.