[dancing spawn of ungoliant]

At least Frodo might have used the name "the Battle of the Peak" in his book when referring Gandalf fighting the balrog. But this is what the Encyclopedia of Arda says about The Battle of the Peak:

(It is) Gandalf's name for his climactic battle with the Balrog known as Durin's Bane, on the peak of the mountain called Celebdil, the Silvertine.

Note: Gandalf's name. So they seem to think that it might have other names as well.