Pat tilapia dry with paper towels; chop. Transfer chopped tilapia to a food processor; add ½ cup mango, ¼ cup cilantro, bread crumbs, green onions, 1 tablespoon blackened seasoning, ½ teaspoon salt, and ¼ teaspoon black pepper. Cover and pulse until coarsely chopped; chill 10 minutes.

Zest and juice lemon and lime in a large bowl. Stir in sour cream, 1 teaspoon blackened seasoning, remaining ½ teaspoon salt, and ¼ teaspoon pepper. Add coleslaw mix, remaining ½ cup mango, remaining ¼ cup cilantro, and sliced jalapeño; toss to coat. Set coleslaw mixture aside.

Divide fish mixture into 4 equal balls. Heat vegetable oil in a large nonstick skillet over medium-high heat. Place balls in skillet and press down with a spatula to flatten to 2 inches thick; sprinkle patties with remaining 1 teaspoon blackened seasoning. Cook 16 to 20 minutes or until patties are golden brown and reach 145°, turning halfway through.

To serve, top toasted bun bottoms with lettuce, fish burgers, cucumber slices, coleslaw mixture, and bun tops.