For me middle name was not an issue. The judge who swore me in (as part of a group) gave a lecture on how the courts don't care what you want to call yourself as long as it's a valid reason.
Name change doesn't take any special documentation unlike gender change. They are unrelated in the courts eyes (but can be done at same or different times of course).
So no, your first, middle and last names have no influence on changes in gender markers.
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