Pour the cooled simple syrup in a large pitcher and add the torn basil leaves. Use a large spoon/ladle to muddle the basil leaves so they release their flavor—beat them up pretty well.
Add the lime juice, rum and Chambord. Stir. Add the club soda, stir, and top with ice if the pitcher allows.
To serve: Garnish each glass with fresh berries and lime slices. Serve over ice or chilled without ice.