I've looked through Reader's Companion, Letters, and History of Middle-earth, and found nothing instructive.
However, even if he didn't literally see the Balrog, I'm inclined to suggest he at least felt it; consider an earlier account of Dáin (emphasis mine):
Then Azog laughed, and he lifted up his head to let forth a great yell of triumph; but the cry died in his throat. For he saw that all his host in the valley was in a rout, and the Dwarves went this way and that slaying as they would, and those that could escape from them were flying south, shrieking as they ran. And hard by all the soldiers of his guard lay dead. He turned and fled back towards the Gate.
Up the steps after him leaped a Dwarf with a red axe. It was Dáin Ironfoot, Náin's son. Right before the doors he caught Azog, and there he slew him, and hewed off his head. [...] [H]ardy and full of wrath as he was, it is said that when he came down from the Gate he looked grey in the face, as one who has felt great fear.
Return of the King Appendix A: "Annals of the Kings and Rulers" III "Durin's Folk"
We can compare Dáin's reaction to the description of the balrog we get in Fellowship (emphasis mine):
Legolas turned and set an arrow to the string, though it was a long shot for his small bow. He drew, but his hand fell, and the arrow slipped to the ground. He gave a cry of dismay and fear. Two great trolls appeared; they bore great slabs of stone, and flung them down to serve as gangways over the fire. But it was not the trolls that had filled the Elf with terror. The ranks of the orcs had opened, and they crowded away, as if they themselves were afraid. Something was coming up behind them. What it was could not be seen: it was like a great shadow, in the middle of which was a dark form, of man-shape maybe, yet greater; and a power and terror seemed to be in it and to go before it.
Fellowship of the Ring Book II Chapter 5: "The Bridge of Khazad-dûm"
Dáin may not have seen the Balrog, and almost certainly wouldn't have known what it was if he had, but it seems exceedingly likely that he perceived the balrog in some way.