I don't think this is really dependent upon setting lore so much as basic tactics. The relevant principle here is "you want to be able to defend as much space as possible via as few points as possible, and you want every one of those points to give its defenders the best advantage they possibly can". If you look at the map:
The Citadel is key to the
overland defense of the area because it means that any ground-based assault originating from within the Karak has to charge up a massive cliff while being bombarded with artillery fire; without taking it they can bombard our overland movement with artillery instead. But notice that it creates a new point of
underground vulnerability, which must be defended against attack from two directions.
Because of this, Kvinn-Wyr is a very attractive next target. Not only does it shorten our line of overland defense as we'll be able to consider the entire eastern valley secure, meaning that Karag Nar, the East Gate, and Karag Lhune all no longer need to fear attack from one direction, taking it will also allow us to effectively shorten our lines of underground defense because we can move the Citadel underground defenses pointed toward Kvinn-Wyr and the East Gate underground defenses pointed toward Kvinn-Wyr both to the Kvinn-Wyr location and point them toward Karag Mhonar. It would do a lot to mark a lot of zones enemy-free. Once that's taken the next expansion wave on that area of the mountain should probably be aimed toward sweeping Grobi Town and as many of the southern peaks as possible, because if we hold the central position then each peak that we take does nothing to increase our underground vulnerability to attack, so taking them purely marks out safe territory. But we'll need a lot more people to occupy those peaks.
Karagril is the closest from Karag Lhune but trying to take it is exceedingly dangerous, because it's got a direct underground line to the orc fortress in the next Karak over
and to our other enemies in other parts of Karak Eight Peaks, and it's as vulnerable to overland assault as much as any giant dwarf mountain is, while being far enough removed from the East Valley area that taking it doesn't actually secure any new territory. Basically, we'd be establishing a fairly conflict-ridden forward position for no serious gains. The only reason to take it next would be if we don't think that we'll be able to protect Und-Uzgar and the associated road to Barak Varr unless we have artillery watching over that section of Death Pass. Arguably, Karagril should be left for the very last thing we conquer in the Karak, since it's potentially going to provoke a fight with a major external force to take it.
