Just note that a passport card is only good for land crossings. If there is an emergency and you need to fly back from an island it won't do you any good.
I could be mistaken, but I believe in this case Princess does require everyone to have passports so there would be no issue in the unlikely event of having to disembark due to an emergency at a port of call.
Yes, Crystal and Fantasy Caves is an attraction that costs admission. To get there, take the ferry to St Georges, then you can take a bus to the caves. You could take a taxi, but it will be a lengthy trip.
I would still recommend the notarized letter. The issue is if he finds out and complains to CBP that you took the kids out of the country without his permission, you'll run into issues when you return.