nah man, even more tsun tsunderes wouldn't promise to make someone's life basically hell
I think they'd try their hardest to ignore the person, not make a scheme to get them to quite school
....now that I think about maybe I got this wrong and she's just a himedere, that would make sense....dang( ̄へ ̄)
Well she is a noble and generally (I’m not saying all of em tho) they do act like that. However Claire does get better later on (based on the novel I’ve read). Also to be fair, she is the villainess from the game so I guess claire inherited all the villainess traits that is expected from the villainess.

Claire's not really supposed to be considered a tsundere, right? 'cause to me she just seems like a bitch
in my opinion a good tsundere is Liselotte from Endo and Kobayashi’s Live Commentary on the Villainess, where her intent is never malicious towards the female lead in the otome game setting, but Claire plainly said she'd bully and harass Rei as if just telling her she was out of line that time wouldn't have sufficed
and as much as I love Rei's disaster gay antics I'd literally rather pair her with any background girl than Clair