This could be an interesting alternative route to get to the desired Bad End...
This girl is ruthless, almost killed a classmate during a duel (Didn't happen only because the MC intervened). and it's not just absurdly strong, but also probably a psychopath.
Rather than trying to follow the normal script and go against the Protagonist to get exiled, why not ally with the Protagonist but direct her towards her worst impulses, becoming a Villanous Duo?
Betray the Prince to side with the Psychothic Commoned that everyone fears (Because you sort of spread that fear), and then cause some sort of incident to make you oficially a traitor and get exiled!