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Eddie Izzard aims to stand in Brighton at next general election
(www.bbc.co.uk)
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Yes, but also has not abandoned her old name either:
source
I actually think I've read that before, now you mention it. I prefer to go with her preferred name and pronoun.
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.