Preheat oven to 350 degrees. Butter and flour a 9-by-13-inch baking dish. In a large bowl, whisk together 1 3/4 cups flour, baking powder, salt, and coconut. In another large bowl, using an electric mixer, beat butter and 1 cup sugar on high, scraping down bowl as needed, until light and fluffy, 4 minutes. Add eggs, one at a time, and beat until combined. With mixer on low, add flour mixture in three additions, alternating with two additions buttermilk, and beat until combined. Stir in lime zest and juice. Pour batter into dish. Smooth top with a small offset or rubber spatula.
In a medium bowl, toss together 1 tablespoon flour, 1 tablespoon sugar, berries, and rum. Scatter berry mixture over batter. Bake until cake is golden at edges and a toothpick inserted in center comes out clean, about 35 minutes. Let cake cool in pan on a wire rack 20 minutes before slicing. Serve warm or at room temperature.
Refrigerate cooled cake, wrapped tightly in plastic, up to 5 days.