The rule for Curse of the Sea states:
At 1st level, when you hit a creature with a cantrip's attack or when a creature fails a saving throw against your cantrip, you can curse the target until the end of your next turn or until you curse a different creature.
Once per turn when you cast a spell, you can trigger the curse if that spell deals cold or lightning damage to the cursed target or forces it to move.
The critical wording here is that both the application and triggering if the curse immediately follow the successful hit or failed save. We know it is immediate because the wording is when not after. So the same hit can't be used both because the triggering requires the curse to be present and the placing of the curse, if it is done, takes up that immediate moment of when the spell is cast.