Whisk 2 tablespoons olive oil and the lemon zest and juice in a large bowl; season with salt and black pepper. Peel the carrots into long ribbons using a vegetable peeler; add to the bowl along with the red onion and half of the chives. Toss well to coat and set aside.
Combine the five-spice powder, cayenne and 1/2 teaspoon each salt and black pepper in a small bowl; rub on the flesh side of the fish fillets. Heat 1/2 tablespoon olive oil in a medium nonstick skillet over medium-high heat. Add 2 fillets, skin-side down, and cook until crisp, about 2 minutes. Gently flip and cook until cooked through, about 2 more minutes. Transfer to a paper towel-lined baking sheet, skin-side down. Repeat with the remaining olive oil and fish.
Add the beets to the bowl with the carrots and toss. Divide the fish among plates; top with the creamy horseradish and the remaining chives. Serve with the carrot salad and lemon wedges.