With her skills as a flatterer, Mrs. Clay is more desirable company for Elizabeth than Anne, because Anne simply cannot lie and say all the things Elizabeth wants to hear. It's not in her nature. Elizbeth has such a weak character that she prefers the company of a "yes man" to her own sister who would probably, if she were ever asked, give her more sensible advice that she is used to hearing. Also Elizabeth is very vain and seems to require almost continuous flattery and Mrs. Clay supplies this need very well.

I don't see this as necessarily inconsistent with her admiration of her cousins in Laura Place. Elizabeth is ironically willing to perform the same type of obsequious toadying that Mrs. Clay does on an hourly basis in order to improve her standing in society. That is just the way the world works in Elizabeth's mind.