The Rape of the Lock is no exception. While Arabella Fermor isn't famous at this point or politically powerful, here Pope has to work his way through a potentially complicated situation: he needs to make sure that Arabella is okay with him taking her story public. And even if she's not okay with it, he needs to make sure his readers know that he at least tried to make her okay with it. He's going to publish it either way.
You might notice that the letter addresses Arabella as "Mrs."; she's not actually married at this point, though. People in the 18th century addressed all respectable women as "Mrs", which was shorthand for "Mistress.