Add lemon juice to one small plate and sugar to another.
Dip the rim of each glass into the lemon juice, then into the sugar to coat. Set aside.
In a cocktail shaker, add the blackberries, 2 tablespoons sugar, and lemon juice. Muddle to release the juices.
Add the Moscato and ice to the shaker. Shake until chilled.
Fill glasses with ice and strain the mixture into each glass, filling about ¾ full.
Top with lemon-lime soda.
Garnish with lemon slices and fresh blackberries. Serve immediately.