Position a rack in the center of the oven and heat the oven to 325°F. Season the snapper all over with salt and pepper. Let sit at room temperature while you prepare the sauce.
Heat 2 Tbs. of the olive oil in a 12-inch ovenproof skillet over mediumlow heat. Add the garlic and cook, stirring, until softened but not golden, about 1 minute. Add the tomatoes and their juice, anchovies, olives, 2 Tbs. of the basil, capers, and pepper flakes to the pan. Bring the sauce to a brisk simmer and cook, stirring occasionally, until the tomatoes are tender and the juices have reduced to a saucy consistency, about 8 minutes.
Nestle the snapper fillets into the sauce, spooning some on top to keep the fish moist. Drizzle with the remaining 1 Tbs. olive oil. Tightly cover the pan with a lid or aluminum foil and braise in the oven until the fish is almost cooked through, 10 to 15 minutes, depending on thickness.
With a slotted spatula, transfer the snapper to 4 shallow serving bowls. If the sauce seems too thin, simmer over medium-high heat until thickened to your liking. Stir the remaining 1 Tbs. basil and the mint and vinegar into the sauce and spoon it over the fish.
Serve with polenta or couscous.