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LW- First, she's well aware of what she's doing. Second, perhaps you should focus less on her behavior (poor) and more on your own. Why would you agree to be in a car for hours with someone who insults you? Why do you want to like this small, mean-spirited person? Why do you need to prove yourself to her? I am not trying to criticize you, just point out some things that would be more productive to think about. It's not realistic to expect to get along with everyone, you need to let this one go. Hoping for friendships with toxic people will only drive you crazy.
If it makes you feel better, the other friends in this group are probably aware of what she's doing as well. Pointing out your degrees and talents will only bring you down to her level and make you look as childish as she is. Think of a witty comeback that will stop her in her tracks, minimize your contact and move on with your life.