Position the rack in the center of the oven and preheat to 350 degrees. Butter two round 8-inch pans with 1 1/4-inch-high sides. Line the bottom of the pans with parchment paper, butter and flour. Combine all the berries in a medium bowl. Add 3 tablespoons powdered sugar and toss gently to coat. Divide the berry mixture equally between the prepared pans, arranging in an even layer. Set aside. Whisk the flour, baking powder, salt and baking soda in another medium bowl to blend. Mix the milk with the lime juice, creating buttermilk, and add vanilla in a small bowl. Using an electric mixer, beat the butter and sugar in a large bowl until light and fluffy. Add the eggs, one at a time, beating well after each addition. Add the flour mixture in three additions, alternately with the buttermilk mixture in two additions, beginning and ending with the flour mixture and beating just until blended. Spoon the batter evenly over the berries in the pans. Bake the cakes until tester inserted into the center of the cakes comes out clean, 40 to 45 minutes. Transfer the cakes to a rack and cool completely in the pans. Invert each cake onto a plate. Remove the parchment paper. Sift powdered sugar generously over each cake. Garnish with a sprinkling of lime zest. Cut the cake into wedges and serve.