Combine sugar, water, and ginger in a small saucepan over high heat. Cook until sugar dissolves, stirring with a whisk. Remove from heat; cover and let stand 15 minutes. Strain; discard solids. Working in batches, place diced watermelon and fresh lime juice in a blender; process until smooth. Strain watermelon mixture through a fine sieve into a pitcher; discard solids. Stir in sugar syrup and chilled club soda. Serve over ice; garnish, if desired.