Cut rind off watermelon and discard. Cut fruit into 1/2-inch pieces. If you want to have garnish for your drinks, salvage some wedges of rind with fruit attached before chopping up the rest.
Line a 10-15-inch pan with plastic wrap. Arrange fruit on wrap in a single layer. Freeze until firm, 1-2 hours. If making this ahead, transfer frozen chunks to a freezer bag.
In a blender, combine half the pineapple juice, rum and lemon juice. Whirl, gradually dropping in half the frozen watermelon chunks, until smoothly pureed. Sweeten to taste with sugar and pour into a pitcher or directly into glasses. Repeat to make a second batch.