So my experience... my family's first cruise was with Passport CARDS -- which, for the record, I do NOT recommend getting simply for the purposes of a cruise, we just happened to have them anyway. This is essentially the same as cruising with birth certificates, because we would not have been able to fly home. We have people with medical needs and we fly from a wintery weather area; we accepted the risks and all was fine. Hundreds of people do this on cruises with no problems.
You say you are "worried about the kids" but I'm not understanding that. The birth certificates get them onto the ship at initial embarkation, put in the safe for the duration, and shown again at Customs and Immigration at final debarkation. The kids only need their KTTW card and will (presumably) be with you when off the ship. Kids younger than 16 need no other ID; kids age 16+ need a government issued photo ID to get on/off the ship just like adults.
Now, some "pros" for the Passports (Books!) if you want to push for that... you don't mention how old the kids are, but Passports are great for LOTS of things once the kids get to school-age and then high school. It's a very simple form of ID, no need to dig out BCs to additionally prove citizenship -- school registrations, camp registration, some sports registrations, part-time jobs for teens, ID for field trips, etc. You'll be on 2 non-US islands, and at-sea -- NO guarantee that any injury/illness at sea will have a guest airlifted to the US, it would be the nearest location. As PP mentions, your DH does not have to be present to apply for their passports, he only needs to sign a notarized form.
Enjoy your cruise!