Put watermelon chunks in a blender and blend until smooth.
Strain the watermelon juice through a fine-mesh strainer into the pitcher.
Add mint leaves to the pitcher and muddle gently to release the oils.
Pour in the rum, lime juice, and simple syrup. Stir well to combine.
Refrigerate for at least 1 hour to chill and let flavors blend.
Just before serving, add ice and top with club soda. Stir gently.
Garnish each glass with mint sprigs, watermelon wedges, and lime slices.
Notes
Keep the club soda separate until serving to maintain carbonation.The base mixture can be made up to 24 hours ahead and stored in the fridge.For a non-alcoholic version, replace rum with extra club soda.Strain pulp completely for the smoothest drink texture.Use fresh mint leaves - dried mint won't give the same bright flavor.