In Chapter 4 Anne still cares about Captain Frederick Wentworth and his marital status as all else about him is of interest to her. Following his career in the navy lists shows that clearly. However, I think you are right and Anne’s thoughts and feelings about Frederick are being described by the narrator and as such is not a script of her thoughts. Anne’s interest in the dear captain bespeaks of hope but I don’t think she is confident he has remained single on her account. As you say there is no evidence for her to know that and I don’t think Anne is so conceited to think so without evidence. Rather, I think the narrator is explaining what Anne knows about Frederick and how she knows it in a humorous and interesting way. Breaking down the sentence:

She had only navy lists and newspapers for her authority, but she could not doubt his being rich; and,

This is straight forward. Anne knows Frederick has become rich through reading of his captures in the navy lists. Then the narrator catches our attention by being less straight forward but enticingly suggestive:

in favour of his constancy, she had no reason to believe him married.