I use firstname.lastname@example.org but attached an address with my name. FaceTime seems to allow selection of the other address but my brother complains about seeing literal appleid@ address
If I understand your situation correctly, you should be able to go to:
Settings > Facetime
...and go down to "Caller ID." This will give you the option of choosing which email address or phone number to display.
EDIT: On an iPhone with iOS 10, that is!