Really, your headcanon has some precedent in the books. If Wormtongue had written the history, he literally would've called Gandalf "bad news." And in fact, Saruman's actual name was Curumo. ...uh, or Curunir. Or Sharkey, or Tarindor, or...

I mean, part of the problem is that every person (and place, and country, and river...) has like a half dozen names depending on who's talking and what time or place they're in. Gandalf himself is Greyhame, Gandalf, Stormcrow, and Lathspell in Rohan alone; and Mithrandir, Olorin, Incanus, and Tharkun to other people in Middle Earth.

Aragorn and Strider and Elessar and Estel and Wingfoot and Longshanks are the same person in different contexts. Galadriel is also Alatariel and Artanis and Nerwen. Legolas is Laicolasse and Greenleaf (all three of which, in fairness, mean the same thing in different languages).

And that's before we even talk about what their names "really" were in the "original" Red Book of Westmarch, before Tolkien "translated" them to English. The "actual" sound that came out of Bilbo's mouth when he introduced himself was Bilba Labingi, but Tolkien decided that the name Labingi "actually" would've sounded like the word for bag or sack to the "original hearers." Likewise Frodo's name is "translated" from Maura Labingi and Sam "actually" introduced himself as Banazir Galpsi.

In my headcanon, that’s not his real name

It's one of his names. As an Ainu (a spirit), he was called Curumo.