I have learned over time that once I explain I am trans one of the first questions to escape the person's mouth is "so what is your birth name," it's something about the mystery that pushes people to wonder for some reason. But most of the time I simply say that I would rather not discuss that topic, and if it doesn't move on to something else, I end the conversation all together.
she needs to learn to respect you for who you are, and if you don't want to talk about your birth name she should understand. if not she is not being a supportive friend anyway.