] Might it be that when Austen wrote "NA", she had only visited Bath, but when she wrote "Persuasion", she had lived there a couple of years?

Exactly, she was 24 when she wrote NA, she saw Bath almost with the same excited eyes of a Catherine Morland, everything there was a new experience for her. But for Persuasion, not only she had lived there, she had lived over all, she was 39-40 years old and she had suffered almost like Anne (lost a parent and love).