I like to imagine Paris One
as having the words PARIS ONE
written across the hull, with the Federation emblem painted in bright, vivid blue directly below it where we're used to seeing the registry number.
If it's analogous to Air Force One, that wouldn't work. Air Force One is not the name of a particular aircraft; it's the call sign of any
USAF aircraft carrying the President of the United States, and it's only called that while the President is aboard. There are two Boeing 747s that are used as Air Force One, but when the President is not aboard them, they use their own individual designations.