So... theres a balrog in the basement...

[Earniel]

If the Dwarves woke the Balrog by delving too deep in the caves, it would make sense that the mines where the Balrog was hidden were a good deal lower than the actual living quarters of Khazad-dûm. I can't see the Balrog sneaking around Khazad-dûm, picking off dwarves left and right, the way Grendel did in Beowulf. So for me that can leaves two scenarios:

If the Balrog took a full year to evict the Dwarves, then it is possible that he did not venture up immediately, staying in the lower levels for the time being. That way his presence wouldn't have disturbed the Dwarves greatly, other than cutting them off from mining. Mithril was vitally important to the Dwarves. In that aspect both Durin and NaÃ*n may have died in battle as they tried to reclaim access to their mines. Dwarves were brave, in an attack both kings would have been up front, and thus quickly killed in confrontation with a Balrog.

If the time between the awakening of the Balrog and death of both kings is shorter, as Valandil suggested, then it is possible the Balrog came to the upper levels fairly quickly after his awakening. To have such a powerful being so close to living quarters would become untenable very quickly. I doubt he would only have killed people, I expect quite a bit of structural damage in his wake as well.

The death of not one, but two kings, would have been greatly demoralising for the Dwarves. I suspect a quick evacuation after the death of NaÃ*n would have occured.

As for Balin's ill-fated colonisation, I would think the Balrog never even knew they were there, or even cared. Moria was a vast space while Balin didn't have such a large colony and I doubt they ventured far from the First Deep. Even the Chamber of Records, where Balin was buried, lay high enough for sun to penetrate.