Place butter, chives, 4 ounces fish, 1 tablespoon mustard, and 1/2 teaspoon each salt and pepper in a food processor; pulse until finely chopped. Add remaining 12 ounces fish to food processor; pulse 5 times or until coarsely chopped. Divide mixture into 4 equal portions; gently shape each portion into a 1/2-inch-thick patty. Sprinkle patties evenly with blackening seasoning; lightly coat with cooking spray.
Whisk together mayonnaise, pickle, parsley, lemon juice, and 1/8 teaspoon salt in a small bowl.
Whisk together oil, lime juice, remaining 1 teaspoon mustard, and remaining 1/8 teaspoon salt in a medium bowl. Add cabbage; toss to coat.
Heat a grill or grill pan over medium-high heat; coat with cooking spray. Grill patties 4 minutes on each side or until done. Remove patties from grill; grill buns, cut sides down, 1 minute or until lightly toasted. Remove buns from grill, and top bun bottoms evenly with patties, tartar sauce, cabbage mixture, and bun tops.