FYI, as this is a closed loop cruise, the Global Entry card would've been fine. Only US Citizens, Green Card holders, and Mexican citizens can get a physical Global Entry card.
Everybody else gets global entry perks but no card (meaning entry via GE by airside only).
Source: my wife is Indian on a visa and enrolled in GE.
To the original poster, I'm glad that you got to your cruise -- but it was a waste of time making you find other documentation, and unfortunately, the person at the "special desk" needs better training. Next time, try to check in with your Global Entry cards as citizenship proof by selecting "SENTRI" upon check-in 🙂
OH -- and for the avid cruisers, be sure to request a 'passport card' on next passport renewal. This will give you a nice card to carry in your wallet that allows for land/sea border crossings.