Heat oven to 340°F (170°C). Line three 8-inch round cake pans with parchment and lightly grease the sides.
Toss the 10 ounces of blackberries with the cornstarch to lightly coat, then refrigerate for 10 minutes to help suspend them in the batter.
In a large bowl, rub the lemon zest into the sugar with your fingertips until the sugar is fragrant and slightly damp, about 30–45 seconds.
Add the eggs to the lemon sugar and beat on medium-high until pale and slightly thickened, 2 minutes. With the mixer on medium, slowly stream in the vegetable oil and melted butter until emulsified.
Whisk in the sour cream, buttermilk, vanilla extract, lemon extract, and 4 tablespoons of fresh lemon juice until smooth.
In a separate bowl, whisk together the flour, baking powder, and salt. Fold the dry ingredients into the wet mixture in two additions just until no dry streaks remain. Let the batter rest for 5 minutes to hydrate.
Gently fold in the chilled, cornstarch-coated blackberries. Divide batter evenly among the pans (about 1 3/4 cups per pan, or weigh for precision).
Bake 24–29 minutes, rotating pans once after 18 minutes, until a toothpick comes out with a few moist crumbs. Cool in pans 10 minutes, then turn out onto racks and cool completely.