Combine 2 1/2 cups blackberries and water in a medium nonaluminum saucepan. Bring to a simmer; cover and cook 15 minutes or until blackberries are very soft. Press blackberry mixture through a sieve over a small bowl, reserving liquid; discard seeds. Combine blackberry liquid, 3 tablespoons sugar, and kirsch; cover and chill for at least 2 hours.
Combine 3 tablespoons sugar and peaches; toss gently to coat. Spoon 1/4 cup blackberry mixture into each of 6 shallow soup bowls. Arrange 1/3 cup peach slices and about 2 1/2 tablespoons blackberries over each serving.