It is possible that a photocopied birth certificate would be sufficient, as the
DCL booklet says. However, it is also possible that even if DCL accepts a photocopy before the cruise, the customs agents you see after the cruise would want to see a certified copy, as the US Customs website implies.
As you said, it is your decision what documentation to bring. I would prefer to carry an original or certified copy and run the risk of needing to pay a small fee to get another if I lose it, rather than run the risk of not being able to board the ship or being delayed after debarkation.