I'm pregnant and Esme is on our list for a girl, said EZ-may. I speak French so it's hard for me to see EZ-mee on Esme just like it's bloody impossible for me to go to dan-YELL for Daniel.
I'm sure there is a bit of confusion about the name and most aren't sure of the proper pronunciation but you'll most often get EZ-may or ES-may (people trying to be fancy and 'correct' without knowing that it should be the hard z sound) because it's not equated with Aimee the way it should be. You'd probably only get EZ-mee out of a French speaker and only if you spelt it in the feminine form Esmee. To get EZ-mee out of Esme in an English-speaking mouth will be an uphill climb and frankly far more trouble than it's worth. Especially if you like both. EZ-may is the usual. You might do a poll to verify that just to explore the avenue fully.
We're not using accents either - we live in England, it's a moot point ;o) Esme Catherine Annuska is probably the front-funner at this stage but I have until Christmas to mess with that ;o)