Izzard will be “remaining Eddie Izzard in public,” she said
Her preferred pronoun is she/her, which the BBC used, and her preferred name for public is Eddie Izzard. I would assume that it is essentially being treated as a stage name for her because of her fame.
If she completely rebranded to Suzy then a lot of people wouldn't know who you were talking about, and she'd lose so much public affiliation. As you say, Eddie Izzard is a stage name as much as it is a brand name.
Eddie Izzard in public, Suzy in private. Makes sense.