I was just on this ship in December. The fomal nights were fairly elegant (especially the first one - the Captain's party). If you eat in the fomal dining room that night, MOST or ALL men will be in at least a suit and you will see a fair amount of dinner jackets (seemed like about 30-40% dinner jackets the first formal night - maybe fewer the second). While I have no doubt there were non-jacket-&-tie wearers somewhere, I never saw any near me - at least none that were over their teens. If he's willing to EVER wear a suit, he should then, or he'll probably look way out of place (at least that first formal night). Interpret that how you wish.
Seriously, you will always find somebody bucking propriety and generally getting away with it, but it's really better if you either dress properly on those nights, or simply choose another dining option those nights. These are very elegant evenings and it's all part of the fantasy - playing along is a lot more fun than party-pooping. (if he still refuses, it's probable, though not assured, that you will be let into the dining room, and it is likely nobody will say anything. But the four other people dressed up at your table will care on the inside!)
You'll love this ship. Gorgeous!