I love this name and hope one day to use it for a girl!
So far as I know,
Zoe and Zoë are both pronounced ZO-ee. The diaeresis does not change the pronunciation. It is merely an accent mark that indicates that
Zoe is pronounced ZO-ee, rather than ZO. This use of the diaeresis is still in use in English, but has been on the wane. Most people omit it; and many people who see the punctuation won't know what it means.
I have never heard
Zoe pronounced Zo-ay. Is that the French pronunciation Zoé?
My personal opinion is that you would be equally correct to render
Zoe with or without the accent marks. (I am myself tempted to write the name Zoë.) However, most Americans ignore accent marks or don't know what they mean. If it will annoy you to see people constantly forgetting to write the name with accent marks ... and some government agencies refusing to record them ... then I'd go with simple
Zoe.