I think the problem with this is that you are assuming that 'they accurately perceive their characteristics to be opposite to their identity.' is not a distortion in the case of transgender people.
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I understand that you believe that the 'inner sense' telling a biological male that their identity is female is valid, accurate and not a distortion.
Millions disagree - identifying as something which is at odds with your biological reality is seen by probably most people as a distorted perception.
I really don't see the the difference between believing that one's body should not have legs, and that one's body should not have male genitalia.
Both are so deeply held by some people that they will undergo surgery to remove healthy but unwanted parts of their bodies. I don't think you can make a distinction between the person who wants to change their body so they have no legs, because they have an inner sense that they should be a wheelchair user, and the person who wants to change their body so they have no penis because they have an inner sense that they should not be a penis-haver.