1Set a fine-mesh strainer over a large bowl and set it aside.
2Place the watermelon in a blender and blend until liquified, about 1 minute. Pour through the strainer into the bowl and scrape a rubber spatula against the inside surface of the strainer to push the juice through until only pulp remains. Discard the pulp. (You should have about 3 cups of juice.)
3Transfer the watermelon juice to a nonreactive 2-quart container. Add the rum, lemon juice, liqueur, and sugar and stir until the sugar has dissolved. Cover and refrigerate until chilled, about 2 hours. Just before serving, stir to recombine. Pour over ice and garnish with a few blackberries.