Perhaps the danger of Mrs. Clay is that she might allow Elizabeth to meet with a man not wholly agreeable to the Baronet.

As I understand the L&T of the period, an unmarried woman had to be accompanied by her parents or a married chaperone when in the public company of unmarried men. I assume that as a widow, Mrs. Clay could attend public gatherings or go to dinner parties at someone's house, unchaperoned. Or could serve as chaperone to Elizabeth in public outings. So, perhaps the implication here is that Elizabeth could go out in public with only Mrs. Clay (no father in sight) and make her own acquaintances with the opposite sex.

If Mrs. Clay is of lower social status, perhaps the danger is that Elizabeth might truly wish to marry someone regarded as unsuitable by her father. Although at her age, she is left with the 35+ group of men, which would be mostly widowers I would imagine, and as we all know, men with status and money have a much larger field of options. And with Elizabeth's airs, I doubt she would try to meet with anyone lower than a baronet herself.

It puzzles me that Sir William has not tried very hard to find husbands for his eldest two dtrs. He and Elizabeth appear to be going to London for part of the season each year. I would presume that the purpose is partly to try to find a match for her, although I suspect that SW doesn't try very hard because he likes the idea of having his eldest dtr running the household. Anne would have no interest in meeting someone else and most likely begs off the yearly London outing. But Elizabeth must be getting desperate by now!