I respectfully disagree with the advice to call the aunt and "explain what is going on." I think that would bring the aunt into the mother-daughter conflict, basically saying "I can't come see you, Aunt, because Mom is being so unreasonable." I can't see that it would help the OP in her relationship with her mother to do that.
My opinion is that the OP tell her mom, "I'm driving to Aunt's in my car" and do so. The OP, while living at home, is an adult, and she can stand up for herself. I don't think that missing out on the weekend on the Cape is necessary. I would take a strong stance here, because otherwise it's going to happen again and again that the mother will pressure the OP to loan her car to her brother, and the OP has very good reasons not to do so.
The trick is to be as polite as possible to the mother. "Mom, I'm sorry, I just can't do that" is a good, vague response when the mother leans on the OP to loan her car. Keep repeating that.