I think a lot of Eloise's anger about Penelope being Lady Whistledown comes down to it being the very last person she expected. Eloise imagined Lady W being this emancipated woman, living off her own riches, poking fun at society from outside of it. That Lady W didn't have to conform to the normal expectations of women, to marry and have babies and obey their husbands. She says it herself in S1, I think it was episode 2 or 3. But for that reality to not be the case, that Lady W was in fact a member of the ton, and someone who did have to conform, and did have to marry was a real blow to this fantasy.
That, of course and the fact that it was Penelope. Of course, she has resentment that Penelope was her best friend and lied to her, and exposed her secrets, especially after Eloise was honest and truthful in their friendship. However, I think it's a bit deeper than that. The awful truth is, I don't believe Eloise thought Penelope was cleverer than her. That's not to say that Eloise believed Penelope to be dumb, but she thought they were the same. That they both held the same beliefs but that Eloise was better and a bigger advocate for those beliefs. And Eloise doesn't like having to confront the fact that Penelope actually does want what many other women are supposed to want, and that Eloise is somewhat unconventional and alone in her outspoken beliefs. And Penelope doesn't want to disappoint Eloise so she does, in many ways, hide her desires, and tries to agree with Eloise's beliefs about womanhood and feminism.
