Bring sugar and 1 cup water to a boil in a small saucepan, stirring to dissolve sugar. Remove from heat, add mint, and let cool completely. Meanwhile, place pineapple in a blender and puree until smooth. Strain mint syrup through a fine-mesh sieve into pineapple puree and pulse to combine. Transfer to a jar, cover, and chill. For each cocktail, muddle 2 pieces of pineapple, 3 lime wedges, and 1 tablespoon mint leaves in a cocktail shaker until lightly crushed. Add 2 ounces rum and 2 ounces pineapple-mint puree to cocktail shaker, fill with ice, cover, and shake until outside of shaker is frosty, about 30 seconds. Pour drink into a highball glass (do not strain) and top off with club soda. Garnish with a lime wedge and serve with a straw. DO AHEAD: Pineapple-mint puree can be made 3 days ahead. Keep chilled.