Preheat oven to 350 degrees, with rack in center. Butter and flour a 9-by-13-inch baking dish; set aside. Prepare an ice bath in a large bowl; set aside. Fill a large saucepan two-thirds full with water; bring to a simmer over medium-high heat. Score the bottom of each peach. Gently put peaches in the simmering water. Using a slotted spoon, remove peaches from water when skin easily peels away from the flesh, 1 to 2 minutes. Plunge into ice bath until cool enough to handle; peel immediately.
Cut each peach into 8 slices. Cut a third of slices in half crosswise (makes about 1 1/2 cups), and set aside. Place the remaining slices in a medium covered bowl, and set aside. In another medium bowl, combine the blackberries and 2 tablespoons sugar. Set aside to macerate, stirring occasionally.
In a small bowl, whip 1 cup cream until stiff peaks form;
set aside for batter. Whip remaining 2 cups cream in a medium
bowl; cover with plastic wrap, and refrigerate.
In the bowl of an electric mixer fitted with the paddle attachment, lightly beat eggs. Add remaining 3/4 cup sugar, and beat until mixture is pale and thick, about 3 minutes. Stir in vanilla. Into a small bowl, sift together flour, baking powder, and salt. Gradually stir flour mixture into egg mixture. Gently fold the reserved whipped cream into the batter in three additions. Pour the batter into prepared dish; spread evenly.
Scatter reserved peach pieces and a third of the blackberries over batter. Bake until the cake is golden and a cake tester inserted in the center comes out clean, about 35 minutes. Transfer to a wire rack, and let cool completely.
To assemble: Cut cake into six pieces. Place three pieces in the bottom of a straight-sided glass serving bowl or compote. Spoon half the fruit (and berry juice) over cake. Spoon half of refrigerated whipped cream over fruit. Repeat with remaining cake, fruit, and cream. Drizzle remaining berry juice over trifle.