Swordfish with Watermelon and Lime-Ginger Citronette

3 tablespoons fresh lime juice (from 2 to 3 limes)

2 teaspoons sugar

2 teaspoons finely grated peeled fresh ginger

3 tablespoons safflower oil, plus more for brushing

Coarse salt and freshly ground pepper

1 pound skinless swordfish steak, cut into 1 ½-inch pieces

4 large scallions, white and pale-green parts separated, dark-green parts thinly sliced

1 head Bibb lettuce (about 8 ounces), leaves separated

½ small seedless watermelon, rind removed, cut into ½-inch-thick wedges

1 medium carrot, peeled into ribbons and julienned

Flaky sea salt, such as Maldon, for serving