Line a baking sheet with parchment paper, arrange the cubed watermelon evenly over it, and freeze for no less than 4 hours or overnight. In a small saucepan set over high heat, stir together the sugar and water until the sugar has completely dissolved; remove from heat and blend in the lemon juice. Place half of the frozen watermelon cubes into a food processor fitted with a steel blade, pour half of the warm simple syrup over the fruit, and puree, pausing as needed to scrape the sides and stir until the texture becomes silky and smooth. Should the puree resist blending due to iciness, drizzle in two tablespoons of water to ease the process. Transfer the finished puree to a loaf pan or a tightly sealed container and freeze. Repeat the same process with the remaining watermelon and syrup. Serve at once for a soft texture or store in the freezer for up to two weeks.