I do it the "other" way--I take our KTTW cards and driver's licenses. In addition, I take a photocopy of the first page of our passports and a copy of the day's Navigator.
I am fully aware that the photocopy of the passport is not a legal document, however, it is useful if I would need to get the passport replaced, and could act in an emergency while the passport is being obtained from the safe on the ship. The Navigator lists the name and number of the Ship's Agent, which is the number you would contact to get a message to the ship in an emergency, to get help, etc.
DCL will facilitate getting your passports out of the safe, luggage, etc. as needed in an emergency.
I was robbed in Europe long ago....I only carry the actual passport where/when I'll need it. I don't need it on any of the Caribbean ports.