Wielding the Flame of Anor

[Eldar14]

Well, this is neither the time nor place to argue the physical size and shape of Balrogs, so I guess I won't discuss the ring not fitting.

But even so, I think Sauron would have been aware of the Balrog's new possession. His eye was constantly searching across ME for primarily the one ring, but also the other rings. The reason he hadn't discovered the three yet was that their wearers had been hiding them, and the Balrog would presumably not do such since he and Sauron were both on the same team.

Of course, this all assumes that Narya could be used by the Balrog. If he was calling upon the Flame of Arnor, then he would presumably be placing upon Narya an essence of the light, which might make the ring painful for use by evil.