It's not about air conditioning. I just feel my typical cruise "nice sundress" with or without a wrap/swawl wouldn't be appropriate for dinner on a Alaska cruise as it's more a summer or Carribean type of clothes.
I'm from Canada and don't wear sundresses in the winter, I don't even wear semi formal dresses in the winter either but that might just be me (except I don't see many people wearing them either, even for Christmas and the like). If going to a fancy restaurant I would wear I heavier fabric with a scarf or jacket (I guess what you could refer to as business attire accessorised for the evening but I was trying to avoid bringing bulky jackets and thinking more scarves).
I've sailed Cunard but it was summer crossings and I had a checked luggage so it was day dresses with a cardigan for the day, cocktail dresses most nights and long formal gown (or more fancy semi formal dress, think designer) on Gala nights.
I'll do a packing test and see what fits the suitcase I guess. I might just repeat outfits by going to alternative dinning so my dinning room table mates are not offended.