Responses
She is the empress's daughter, switch to ensure she got the crown, so she is the prince's sister. Also the dad hates her because his wife and real child died during the switch. It just sounded so astronomically stupid and a rather over the top 'twist' I got 'iris lady with a not so smarrphone' vibes.

If Lesian (Why She Lives as a Villainess) were in Dahlia's place she would have risked or sacrificed her life like a badass bitch to humiliate and cut down the scum emperor and his men.
If it were Melissa (Beware of The Villainess)...She very likely would have done the same and also find any ways to become so powerful that she can viciously + unapologetically destroy any opponents.
Wishful thinking