The series portrays Olivia/Madame Lulu as a misguided and tragic character who's philosophy is to give people what they want. Putting her betrayal of the Baudelaires aside, this portrayal ignores the fact that she shows just as much cruelty as Olaf, or at least in regard to her employees. She refuses to pay the freaks, never gives them presents, and denied them more time to rest even though they would be more effective well rested as one of them pointed out. If her motto is to give people what they want and the freaks wanted money and good rest (They may not have cared about getting presents until Esme has given them some), isn't that hypocritical and cherry picking who she gives to?
Also, she completely approved the Lions Pit and was willing to have the lions eat a freak. The only thing she objected to was throwing the Baudelaires down (and if she never found out about their true identities and thought they were real freaks, she wouldn't have.)
I can buy that she might have been remorseful with exposing the Baudelairs to Olaf but she had no such regrets doing the other cruel stuff. Is it really this surprising that the real freaks would agree with Esme in betraying and killing Lulu and join Olaf's troupe?
Madame Lulu is very important in my theory. She also surprises me. It presents the concept of Schism. But she says she does not remember several details. The Great Schism apparently hit Lulu in the face. She, like the Windershins family, seems to believe that there is a bit of logic in the actions of the incendiary side. Likewise, she must think there is a bit of evil in the actions of the fire-extinguishing side. But she herself wants to be a good person. How to deal with all this? How do you try to stay neutral in this war? That's how I understand Olivia's way of thinking. She is a person tormented by her beliefs. I would say her fundamental belief is, "The world needs a secret oraganization looking out for it." In this respect, Olivia is reproducing what she has learned, probably since childhood. A person fighting against their beliefs is very difficult. It succeeds in being subjected not to a leader but to a concept. She says to herself, "I am subject to a very important and structured secret organization." She says to herself, "Even though I do not understand what the organization is doing, or which side I am supporting, I have to submit to organization, because the world needs a secret organization to look out for it." In other words, Olivia is exactly the opposite of Olaf. Olaf is the Kaos tool.
I think it's shown throughout TCC that, while Olivia may favour giving people what they want, she has biases about who to prioritise. She prioritises the entertainment of the public over the freaks' welfare; and she prioritises Count Olaf over the Baudelaires. She's spent so long in her disguised role that she's perhaps lost touch with the majority of her human feelings; the only personal feeling she doesn't disguise is her infatuation with Olaf - who appears to know her only in her disguised role, for all that they must have been volunteers at the same time.
I think you are right about her being hypocritical. I think in her mind, the freaks are not people (which is abhorrent to us), and this is how she justifies not treating them like people.
I guess when you put it that way, it is not surprising that the freaks were willing to push Lulu into the lion pit, although I would say murder is a bit worse than forcing freaks to put on a show, but on the third hand, Lulu was okay with a freak being pushed to the lions.
The thing I still don't understand is what Dante touched on - Why is Lulu infatuated with Olaf?
Faith is being sure about what we hope for, being convinced about things we do not see. Hebrews 11:1 For God so loved the world that he gave his only-begotten Son, that whoever believes in him shall not perish, but have eternal life. John 3:16