Cook butter, oil, garlic, anchovies, ½ teaspoon of the salt, and ¼ teaspoon of the pepper in a saucepan over medium-low, stirring often, until butter is melted and anchovies have mostly dissolved, 4 to 6 minutes. Set butter mixture aside.
Preheat a grill to medium-high (400°F to 450°F). Toss eggplant, okra, onion, 2 tablespoons butter mixture, ½ teaspoon of the salt, and ¼ teaspoon of the pepper in a large bowl. Place seasoned vegetables on oiled grates. Grill, turning occasionally, until lightly charred and tender, 3 to 5 minutes for okra, 8 to 12 minutes for eggplant, and 14 to 18 minutes for onions.
If remaining butter mixture has solidified, reheat in a small saucepan over low, stirring often, until melted.
Remove from heat; whisk in vinegar. Arrange tomatoes, eggplant, okra, and onions on a serving platter.
Sprinkle with remaining ¼ teaspoon each salt and pepper. Drizzle with butter mixture. Garnish with purple basil and radishes.