Cut the watermelon in half and use a spoon to scoop the flesh out into a blender or food processor. Work in batches as necessary so as not to overfill your appliance. Process the watermelon until smooth, then transfer to a clean kitchen towel.
Squeeze the juice through the towel into a large pitcher and repeat as necessary until you have the juice of the entire watermelon. Once you have all the watermelon juice, add the juice of the two limes and stir.
To assemble the mocktails, fill a glass with ice. Pour watermelon juice about one third of the way up the glass. Take a few mint leaves in your hand and give it a good slap with your other hand to release the aroma. Add the mint leaves to the glass and top with club soda.