DS was born in 2009 and a woman who was with me at the parent and birth classes at the hospital named her daughter Esmee. It's been getting more popular for several years, even before you had your DD.
I don't think you can change your daughters name now just because you're upset that someone else has the same name.
I'm sorry I can't think of a more gentle way to say this but if you are really "hacked off and upset" to the point that you would change a four year old's name because you don't want someone else in her class to share it, and you're basing this on your own experience and still shudder at hearing your own 'real' name years later, I think you might benefit from talking this through with a counsellor.
This seems like you have a deep seated issue about names that you might feel better for unpicking.
What is it about sharing a name that bothers you to this extreme?
And with the best will in the world, whatever it is, it's YOUR issue. You say you don't want to put your children what you went through, but all you went through was having four girls in your class share your name.
Your daughter might not mind having others share her name, but if you make her believe there's something wrong with that you might actually be the cause of her having a problem.
It's not fair to put your feelings onto her and make her feel like there's something wrong with her name, and asking her if she wants to change it is putting your issue on her.
What will you do if she chooses a new name and then in another year or so another child with the same name turns up? She can't keep changing her name to be unique to make you happy.