Não claro
Desculpe.
Eu olhei através Companheiro do leitor, Cartase História da Terra-média, e não encontrou nada instrutivo.
No entanto, mesmo que ele não tenha literalmente Vejo o Balrog, estou inclinado a sugerir que ele pelo menos erro isto; considere um relato anterior de Dáin (grifo meu):
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"
Podemos comparar a reação de Dáin com a descrição do balrog que encontramos Amizade (ênfase minha):
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 pode não ter visto o Balrog, e quase certamente não saberia o que era se tivesse, mas parece extremamente provável que ele tenha percebido o Balrog em alguns caminho.