She is apparently such a good toady-er that she is beyond suspicion. In chapter 5 when Anne tries to warn her sister, Elizabeth replies, "Mrs. Clay never forgets who she is, and as I am rather better acquainted with her sentiments than you can be, I can assure you that upon the subject of marriage they are particularly nice; and that she reprobates all inequality of condition and rank more strongly than most people." She goes on to explain why her father is in no danger, having kept himself single, and how unattractive Mrs. Clay is and is therefore no threat.

I agree with Maisy that Elizabeth isn't very deep, she trusts Mrs. Clay because Mrs. Clay knows how to say "the right things" and this establishes her as being trustworthy.