Yeppers.....you should be good to go now and you do not need to take the birth certificate off the ship when in Bermuda...just your boarding card and photo ID for anyone 16 years and older...under 16 years of age, no photo ID needed to debark or embark ship while in Bermida, but you may want to bring that notarized document with you when leaving the ship, just in case you are asked for it.
Have a GREAT cruise !
PS... looks like the minor child has a passport and that "trumps" a BC and BC is really not necessary at all. Some will encourage you to carry passports off the ship, some will say bring a copy of passport when off the ship in a port of call..... really your call on this. I would just follow what the cruiseline suggest in your daily paperwork provided to you with suggestions and requirements of what is needed when debarking ship.