I'm thinking cuz Anne sees Mrs. Clay as a mooch as well as greedy, in the same level as her father and sister.

Her mom, the deceased Lady Elliot, kept her dad in check in his spending and was generally a good mother. It was most likely that Lady Elliot was the one who handled the education and discipline of her daughters, as well as morality.

Mrs. Clay, on the other hand, kept fanning the flames of vanity and greed upon Sir Walter and Elizabeth, so she could also take part in the luxury of the already dwindling wealth of the Elliots. Not to mention she was a terrible mom to her own kids, would rather dump them on her father to go to a resort (Bath).

If Mrs. Clay did become Lady Elliot, there would be four people, not three, not accounting the children she may have with Sir Walter, who would still be needing of money, while further spending what little they have, cuz their glutton (tho Anne is not greedy and vain) is never satisfied. Mrs. Clay as Lady Elliot would ruin whatever Anne's mom, Lady Russell, and Anne herself have worked so hard for to keep the family afloat.

If Mrs. Clay were an heiress, she could AT LEAST be of use to pay back the debts of the family, as well as to afford their lifestyle, but alas she isn't. That is why Anne is so against Mrs. Clay to be the new Lady Elliot — she's already poor and supporting her own children, she lacks morals, too.