Make panna cottas: Purée blackberries with buttermilk in a blender until very smooth, then pour through a fine-mesh sieve into a bowl, pressing on and then discarding solids.
Sprinkle gelatin over water in a small bowl and let stand 1 minute to soften.
While gelatin softens, heat cream and sugar in a saucepan over moderate heat until hot, stirring until sugar is dissolved. Remove from heat and add gelatin mixture, stirring until dissolved. Stir cream mixture and syrup into blackberry purée, then pour through cleaned sieve into a bowl. Pour mixture into molds and chill, covered, until firm, at least 8 hours.
Make compote while panna cottas finish chilling: Boil water, crème de cassis, and sugar in a small saucepan, stirring occasionally, until syrupy and reduced to about 1/3 cup, 10 to 12 minutes. Stir in lemon juice, then pour syrup over blackberries and gently stir to combine.
To serve panna cottas, run a thin knife along edge of molds to loosen if necessary and dip molds in a small bowl of warm water 3 to 5 seconds, then invert panna cottas onto plates and gently lift off molds. Spoon berries and syrup over and around each panna cotta.
Cooks' notes: *To make blackberry syrup, heat 1/4 cup blackberry jam with 1 tablespoon water in a small saucepan over moderately low heat, stirring, until jam is dissolved. Pour mixture through a fine-mesh sieve into a small bowl, pressing on and then discarding solids. *Panna cottas can be chilled in molds up to 1 day. *Compote can be made 2 hours ahead and kept at room temperature.