In a saucepan on the stove, continuously stir and heat the sugar and water until sugar has dissolved (the mixture will be clear with a slightly golden tint). Turn off the heat, and set aside to cool (put it in the freezer if serving soon).
In your blender, combine watermelon and 15 oz (just shy of 2 cups) lemon juice. Zest one large lemon into the mixture. Juice your naked lemon into the blender.
Blend the mixture until it’s smooth and combined (about a minute). Place the strainer over the bowl, and pour the watermelon mixture through the strainer. Use a spoon to stir the mixture around in the strainer and ensure all the lemony goodness makes it into the bowl.
Combine the watermelon-lemon juice mixture and the sugar water into a pitcher and stir to combine. Place in the fridge to chill (at least an hour but perferably overnight).
For the Punch
In a large punch bowl, pour in the lemonade, then the rum. Stir well with a wooden spoon.
Pour in the club soda a bit at a time (to prevent fizzing over).
Top with frozen watermelon to stay chilled. Drink immediately!