Puree the blackberries, lychee syrup and lemon juice in a blender. Strain through a fine-mesh sieve into a pitcher, pressing on the solids with a rubber spatula. Add the acai juice and blueberry juice. Refrigerate until chilled, at least 2 hours.
Gently push a blueberry into the cavity of each lychee. Transfer the punch to a punch bowl or pitcher and add the stuffed lychees, then gently stir in the lemon-lime soda and rum, if using. Serve over ice.