Sprinkle the tilapia on both sides with half of the salt and the pepper. Heat the oil in a large skillet with an ovenproof handle over medium-high heat until it begins to shimmer but is not smoking. Place the fillets skin side up in the hot oil and sear until the fish begins to turn color, 1 to 2 minutes. Turn the fillets and add the lemon juice, wine, and butter to the pan. As soon as the butter melts and becomes foamy, add the capers and tomatoes and simmer until the tomatoes release some juices, about 1 minute.
Place the pan in the oven until the fish is cooked through, 5-7 minutes. Baste the fish with some of the juices and transfer to serving plates. Add the cup torn basil and the remaining salt to the pan juices and stir until the basil is slightly wilted. Spoon the mixture over the fish, garnish with fresh basil leaves, and serve.