in this case choosing to cruise during a school break and NYE at that is a mistake. Try a week later or the first two weeks of December - prices may be a lot cheaper and fewer kids as well.
By ship, I'd choose either Apex or Royal Princess.
By itinerary, Apex and RP have lovely destinations. We love both ABC islands (Southern Caribbean) and Eastern Caribbean itineraries. However, we love beach destinations now that we travel with a somewhat young child. HAL's itinerary is a very common Western Caribbean.
By price, I'd choose Royal Princess, but I would try to get a Deluxe balcony instead, because it would provide a sofa which becomes the 3rd bed. Or HAL because it's over NYE and for only $4.8K for 3.
Either way, all three ships/lines provide a great experience for older, quieter crowd. Since your child is only 9 years younger than me, I would estimate you to be between 55 and 70...