I’ve never thought that the Balrog had wings. It was only a figure of speech.
There was a darkness or ‘dark fire’ around it, which was compared to wings spread out and blocking the light around it.
Descriptions:
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.
It came to the edge of the fire and the light faded as if a cloud had bent over it.
The dark figure streaming with fire raced towards them.
For a moment the orcs quailed and the fiery shadow halted. Then the echoes died as suddenly as a flame blown out by a dark wind, and the enemy advanced again.
His enemy halted again, facing him, and the shadow about it reached out like two vast wings.
“The dark fire will not avail you, flame of Udûn. Go back to the Shadow!”
It stepped forward slowly on to the bridge, and suddenly it drew itself up to a great height, and its wings [of shadow] were spread from wall to wall
With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished. But even as it fell it swung its whip, and the thongs lashed and curled about the wizard’s knees, dragging him to the brink.
Tolkien says the shadow was LIKE wings. If it actually had wings, he certainly wouldn’t say that. He wouldn’t use a simile. He was describing what the shadow was like.
The second time he mentions ‘wings’, the fact that they are dark shadows like wings is to be understood.
He also used two other similes to describe the phenomenon. It causes the light to fade ‘as if’ a cloud had bent over it, and ‘as a flame blown out by a dark wind’.
The darkness or shadow around the Balrog is compared to a cloud, a dark wind, and wings. But ‘what it was could not be seen’. The wings are not literal.