Some thoughts
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IMO the best way I can see understand this and the overall trans fixation on 'you're erasing my existence' is to look at it in terms of the theory that 'Lisa' is a mental structure produced via dissociation and compartmentalisation - a kind of mind-within-a-mind.
If this were true, then Richard Dawkins is literally threatening the existence of 'Lisa'. Any info which destabilises the beliefs and behaviours that hold the mind-within-a-mind (dissociative identity) together actually *DOES* threaten the 'existence' of 'Lisa'. It threatens the sleeping over-mind's ability to continue to believe that it is Lisa and nothing more than Lisa.
'Lisa' may have started existence as an autogynephilic sexual fantasy. This man may have gained comfort by deliberately acting out the part of Lisa via crossdressing etc. Over time, however, this habit was entrenched, so that he acted like 'Lisa' without realising it. This would likely have entrenched the dissociation and compartmentalisation.
Similarly, in schizophrenia, superficially nonsensical or enigmatic statements have actually been recognised to be oblique attempts to communicate things going on in the person's head. I would recommend R.D. Laing's book 'The Divided Self' if you want to learn more.