I am forming an opinion that Mrs Clay as companion to Elizabeth was possibly ousting Anne from her rightful place in the family.
Elizabeth did prefer her 'agreeable' company to Anne's.
I agree with a point you made that in Bath with the Elliots, Mrs Clay as Elizabeth's friend and hanger-on, can share their genteel lifestyle. I think flattery makes Elizabeth quite malleable to the influence of the appropriately named Mrs Clay.
As we know, Lady Russell regards Mrs Clay as a dangerous companion yet I wonder if this is partly motivated by snobbery. Lady Russell knows the Elliots are already in danger of losing rank and consequence; she must deplore their intimacy with a woman of the lower class.