There's no such thing as a dumb idea.
It probably wouldn't be possible for the player to manually select a destination, but using a 'Sail To' style option would work.
Perhaps we could disable Sail To for all ships except the Dutchman, even if we can't make it submerge and re-emerge properly.
I know we can change the waterline of a ship, potentially even at sea, so it's possible to submerge the Dutchman in theory.
However, the whole thing might not look very good, because the ship would simply lower vertically, rather than actually diving nose first.
Although, I have seen a way that ships can be forced to capsize, which obviously involves changing the side-to-side roll angle, so perhaps the forward-backward angle could also be adjusted in a similar way.
If that were combined with simultaneously lowering the ship's waterline, it might simulate diving. I'm not sure how easy that would be, but in theory it might be possible.