Having her respond by challenging the other girl is a better reaction, but I don't think I'd feel sorry for Ariana for doing it. It'd just be within her character. Not sure if this would be a clear enough example, but it'd be like feeling sorry for Ash because he's friendly or for Bill because he's intelligent. Ariana is proud, self-confident, and assertive, so she would respond by attempting to prove the other girl wrong.
Remember, another part of making readers feel sorry for a character is answering the question of why we should feel sorry for them. That's done by establishing a lot of lead-up to the moment where Ariana snaps or challenges someone, and it's why just focusing on one moment where a character faces a problem and lashes out doesn't always work. That's another reason why that fic where Palkia is hoping Cresselia gets hurt doesn't work, actually: because although we're told Cresselia bullied Palkia a lot, we're not really given a sense of how horrible life was for Palkia during that time. If we were given a lot of snippets and scenes in which Cresselia and others pushed Palkia around and in which Palkia couldn't respond, it'd be easier to feel sorry for Palkia because we'd get to see how frequently she got pushed around. That would have given us more time to develop emotions for the character and to root for them. Sure, responding to bullying by writing a lengthy essay about how you wish your bully will get hurt is still inappropriate, but at least we'd understand why it came about a little better if we were given plenty of scenes of the bullying taking place.
Tl;dr, I probably wouldn't feel sorry for Ariana if all I got was a scene of her challenging a girl after a few insults, even if those insults were pretty serious. It's seeing (through actual scenes) a ton of abuse leading up to that moment that would make me feel sorry for her and root for her.