That’s precisely the point, in my humble opinion. I do not think the young lady’s familiarity with Shas is too great, and I also notice that she is asking a woman, she did not ask her father / brothers / Rabbi (or she asked and they did not answer; possibly there is a reason for that). I think you might tell her you asked and were given this answer (print it out): if she understand it, awesome, she will have her answer (BTW thanks Popa for sharing this beautiful insight). If, as I suspect, she can’t, she will be forced to reach the correct conclusion, ie that she is ignorant – and that’s the very reason she asked the original question. Yet, if the question was sincere, she will go to her father or brother or Rabbi and ask them to please explain what does it mean, admitting that the various cases when the halacha is according to Rabbi Abaye and someone who lost hope about a lost object and a mumar etc. don’t seem to her to have a relation with her question, and will ask them to please explain why is this supposed to be an answer to a question of hers – which she will respectfully state. In the event the question was insincere, now it is her problem, you have quoted her from the Talmud, which to a woman is unusual and yet you promptly did, and there is nothing more she can ask for.
If you are her teacher, I find her statement you quote “As a ‘feminist’ I believe it is the men’s sacred duty to look up interesting Midrashim, meforshim and halachos for me, answer all my random hashkafic questions, and tell fascinating Divrei Torah. In return, I have to keep tznius.” if that is quoted correctly, quite disrespectful and not the proper way to address anyone, let alone a teacher.