In a blender, in batches, puree watermelon. Pour through a fine-mesh sieve into a liquid-measuring cup, pressing on solids to extract as much liquid as possible (about 3 cups); discard solids.
In a large container, combine mint, sugar, and lime. With the back of a wooden spoon, crush mint and lime until sugar is dissolved. Add watermelon juice and rum, fill container with ice, and stir to combine. Serve over more ice with mint sprigs.