Hi cameron, and welcome to cruise-chat.com.
I can't answer the question on the private island as I have yet to visit. However, having taken our son on almost all of our cruises since he was a year old, I would definitely recommend taking them to dinner with you on the formal nights. It will be easier on all of you if you have early seating or My Time dining, as the late seating may be too late for them.
When our son was little, we used to get a table just for our family. But, Royal Caribbean is great at putting families of similar ages together for dining.
To get them used to dining, I recommend starting out slowly at home -- first going to very casual restaurants near you, and eventually working up to those in which the meal takes longer to be brought to you.
Of course, if you want to have an evening to yourselves, you could have them eat in the kids club on one of these nights (most lines will do this -- I leave this to the RCCL experts to confirm).
They might also enjoy the evening shows. My son did. When he was that age, we'd sit more toward the back of the theatre in case we needed to make a quick exit. (Which was rarely necessary.)