Whisk soy sauce, rice wine, 1 tablespoon olive oil, hoisin sauce, fish sauce, oyster sauce, and juice of 1/2 the lime together in a bowl.
Heat 3 tablespoons olive oil in a skillet over medium heat. Cook and stir shallot and ginger until starting to brown, about 2 minutes. Pour in soy sauce mixture; simmer until slightly thickened, 5 to 10 minutes.
Score both sides of the pompano in a crosshatch pattern. Pat dry with a paper towel. Season both sides and inside of the pompano with dill, salt, and pepper. Slice remaining 1/2 of the lime and arrange slices inside. Drizzle remaining 1 tablespoon olive oil over pompano; rub over both sides.
Layer cabbage leaves in the bottom of a large baking dish; place pompano on top. Pour soy sauce mixture over pompano. Cover baking dish with aluminum foil. Let marinate as oven preheats.
Preheat oven to 400 degrees F (200 degrees C).
Bake pompano in the preheated oven until fragrant, about 20 minutes. Remove aluminum foil; continue baking until pompano flakes easily with a fork, about 10 minutes more.
Peel back skin and lift pompano flesh off the bones using a fork. Transfer to a serving plate.
