The question breaks down into two parts: Why a swordsman instead of an axeman, and why the swordsman of Calais.
I now believe that the answer I posted earlier was only partially correct. This source offers a more likely answer, that "Henry did not care about Anne's feelings," and that he chose the sword as "a"the symbol of Camelot and the, of a rightful king."
On the other hand, Anne didn't seem to mind the substitution. According to the article"
"‘I heard say the executioner was very good and I have but a little neck,’ Anne said the day before her execution, and laughing, she put her hands round her throat. It was, at least, to be a quick death: her head fell with one blow, her eyes and lips still moving as it landed on the straw."
Original answer:
Anne Boleyn's PREFERENCE was for a "swordsman" rather than an "axeman." (This is different from my earlier, and now "retracted" statement that it was her request.) In the letter referenced in the question, the warden of the Tower of London reported that: "I have seen many men and also women executed, and that they have been in great sorrow, and to my knowledge this lady has much joy in death."
The initiative appears to have come from Henry VIII, who seemed to anticipated that the swordsman would provide a less painful, and hence "preferred" death (unless they had discussed this beforehand, perhaps in another context). If so, he is a better man than I have given him credit for.
That may be because she could then die kneeling upright, instead of with her head on the block.
The Calais swordsman was chosen as the "best in class."
Anne Boleyn was treated with more consideration than Mary Queen of Scots, who died with her head on the block, requiring TWO strokes of an axe, and also Catherine Howard, whom Henry also executed.