I have first hand experience with this on Martinique
. They require original documents and are smart enough to know if you are showing the genuine document. At that point they will also ask you the direct question if it is the original or a copy? Would you lie in that case? I didn't and went to get originals.
I have never heard of any East Caribbean
island taking documentation
hostage, except when the boat owner is suspected of criminal acts (incl. not paying their bills).