The BBC used 'living as a female', and other people are using the phrase 'living as a woman'. Frankly, I would have slightly fewer problems with using the latter -- you can after all be male, and attempt living as a woman (i.e. if you pass well enough then you might in some social situations get treated as if you were female). But then this murderer would fail at that as well, since no one is going to think for an instant that they are female, hence although someone might recognise that because of their clothing, name etc they are attempting to 'live as a woman', they'll get treated as a transwoman, rather than as a woman.
Frankly, 'living as a transwoman' would be a more accurate description, since that would correctly mean being male, but 'presenting' as something else, whether or not you think that is valid or possible or desirable.
So why is that so unacceptable a phrase? It's true, accurate, and doesn't step on anyone elses toes, and transwomen can argue amongst themselves what it should or does mean, and leave the rest of us the hell alone...............