They are concerned about sea sickness, not going on a liveaboard without their dive shop.
Sea Sickness in all in your head. Of course, when it's in your head and you spend 4 of your 7 days barfing, it doesn't matter that some guy on the internet said it's psychosomatic, you're still barfing. I have found the best preventative is the Transderm Scop patch. Be aware, don't just stick it on and think you've done enough. It will come off while diving, and you will get sick. Also, don't put it on and rub your eyes, or you may blow a pupil for a week, making you extremely sensitive to sunlight. Also not a fun way to spend a week. Put the patch on after cleaning the area behind your ear with alcohol to remove any skin oil. Cover the parch with a good waterproof bandaid. Wash your hands. Wash them again. The patch is good for about 4 days, I'd change it on day 3 for the second half of the trip. A prescription comes in 4 packs, so you will each have one to play with at home to be sure you don't have a bad reaction, one to lose on the first dive, and 2 to get through the week with.
Have fun in Saba/St. Kitts. I used to work on the predecessor to that boat, I love the diving there. You are in the lee of the islands, so it isn't like you're facing the waves and wind all day every day, and the crossings are at night. I get seasick too, although my time on the Caribbean Explorer passed without turning into a barf machine myself.