That description of Westgate Buildings is Sir Walter's - a not exactly reliable opinion. Westgate Buildings was one of the places the Austen family looked for lodgings when they moved to Bath, so I hardly think it's a slum.

I was never surprised at Lady Russell approving of Anne's visiting Mrs. Smith. She knows Anne is miserable in Bath and, even if she doesn't know the extent to which Mrs. Smith helped Anne while at school, I'd think she'd support anything that might strengthen her ties to or increase her enjoyment of Bath.