Some might think that using any sort of accents implies pompousness. There is also the fact that the French Chloé has a different pronunciation - klo-AY as opposed to klo-EE.
I might recommend not having any sort of accent. Few people use it, many are confused by it, and it will not show up in many documents - no matter how important (as a Chloë myself, I can attest to it!). If you feel like adding an accent, you can every now and then.
Hello and welcome, other new
Chloe! :D
PS: How long have you thought about changing your name for?