Responses
I think because Chloe awakened, Chloe would get attention and love from the Duke. Blaire prob wanted the Duke to treat Chloe well. If Chloe kicks up a fuss about Blaire, she might lose the Duke's favor.
Blaire kinda explained it. Claude already hated the Duke, that's why Blaire thought that if Chloe at the time knew that she got kicked out by the Duke, Chloe/Claude will hate the Duke more. And since in the original story that Blaire remembers, Chloe is not that completely angry with the Duke after Chloe had her awakening.
That's why I didn't understand why she wanted to keep the fact that she got fired hidden in the first place. It would only hurt chloe more to let her think that blair left because she wanted to rather than against her own will. And now she throws it back at Claude like it's his fault for misunderstanding. Sigh.
I'm glad it's cleared up though.