Yep, my son's last name is different from mine. No one batted an eye at us when we disembarked in Nassau (although I did have his passport, just in case). Our other stops were Key West and
Castaway Cay, so there were definitely no issues there.
The only potential issue I could see is if you book port excursions and then are denied entry because you don't have the letter. You would likely lose whatever you paid for the excursion. I really don't think they scrutinize when a child is with only one parent because you are not going to be able to STAY in the country - you have to get back on the ship before it leaves port. Compare that to if you were flying or driving in, then there could be the potential for a parental abduction (which is what all this documentation is trying to prevent). It just doesn't seem logical that a parent would use a cruise as a method for leaving the country with their child permanently...even more so since you DON'T have a passport at all for your son.
I definitely wouldn't cancel...if you are really worried, there is PLENTY of fun to be had on the ship, and not going into port will probably save you money in the long run