I'm aware of the fact, that "fly" can mean "flee". But can you prove that Tolkien meant "flee" in this special case? That's what I doubt, not that it can mean "flee".
Well yes, as I said the fact that Tolkien frequently used
flight in that sense seemed like a pretty good proof to me. I guess you could also look up how this specific passage was translated in a version where the translator worked closely together with Tolkien (like the german Carroux Translation for example, unfortunatly I can't look mine up atm). I suppose it is not too farfetched that this passage would also have been noticed by a translator.
But I think you're looking for proof in form of a defintive answer from Tolkien, like a letter or the like. That I guess you already know that there is none.
So okay, let's look at it from another angle. Let's assume Gandalfs perspective during his confrontation for a moment:
Gandalf who is desperately trying to get Frodo (and the Fellowship) unharmed through Moria. At the last moment the Balrog catches up with them. Gandalf knows that he is the only one powerful enough to do anything against this foe, but even this might not be enough. He is trying to stop him at the bridge, opposes him with his magic ("the dark fire will not avail you!" after which the fire of the Balrog does dwindle for a moment), and even deflects one direct blow (breaking the blade of fire).
It is at that point that Gandalf decides:
"I will break the bridge under him, so that we might be able to flee."This makes
no sense if the Balrog could fly. Why should Gandalf even try this? Why would he assume that breaking the bridge will throw the Balrog off balance? The Balrog would catch up with them just a moment later. It would be annoying sure, but not really effective. This would make Gandalf pretty stupid for even trying something that would obviously be ineffectual.
But maybe Gandalf planned to wrestle the Balrog in midair and thus hinder him from flying!That makes
even less sense! First remember that the Balrog pulled Gandalf down with his whip. Gandalf did not plan to fall with the Balrog, he didn't used the bridge as spring board, deliberatly jumping into the chasm. So even if he could have prevented the Balrog to just fly back up, it was definetly not something he did by choice.
Second,
could Gandalf honestly prevented the Balrog from flying? I mean yeah, the scene in the movies looks great and all and Gandalf is pretty powerful, but how exactly should that work? The winged Balrog could have regained his balance in midair and dodged Gandalf, because I suppose Gandalf can't really (quickly) change the direction in which he falls. In the movies they were both
just falling. Otherwise, Gandalf would have to jump onto the flying Balrog and somehow keep hanging on him, which I'm pretty sure the Balrog would do anything against.
And this is Gandalfs Plan to stop the winged Balrog? As I said, this doesn't really paint Gandalf as a competent, foresightful character. (Which as the rest of the story shows us is
exactly what he's supposed to be!)
That's why I'm pretty sure that this age old question can be with reasonable certainty answered. Of course you can choose to interpret the few passages als literally descriptive, but by doing so you're missing a lot of context.