Plan ahead: The pastry cream needs to chill for 3 hours so it will set before you assemble the cake.
1. FOR THE CAKE: Adjust oven rack to lower-middle position and heat oven to 350 degrees. Grease three 8-inch round cake pans, line with parchment paper, grease parchment, and flour pans. Whisk flour, 1¼ cups sugar, baking powder, baking soda, and salt together in large bowl. Whisk buttermilk, egg yolks, melted butter, oil, and vanilla together in second bowl. Add buttermilk mixture to flour mixture and whisk until just combined.
2. Using stand mixer fitted with whisk attachment, whip egg whites and cream of tartar on medium-low speed until foamy, about 1 minute. Increase speed to medium-high and whip whites to soft, billowy mounds, about 1 minute. Gradually add remaining ¼ cup sugar and whip until glossy, stiff peaks form, 2 to 3 minutes.
3. Whisk one-third of whites into batter. Using rubber spatula, gently fold remaining whites into batter until no white streaks remain. Divide batter evenly among prepared pans, smooth tops, and gently tap pans on counter to release air bubbles.
4. Bake until tops are light golden and toothpick inserted in center comes out clean, 18 to 22 minutes, rotating pans halfway through baking. Let cakes cool in pans for 15 minutes. Remove cakes from pans, discarding parchment, and let cool completely on wire rack, about 2 hours.
5. FOR THE PASTRY CREAM: Whisk egg yolks, 2 tablespoons sugar, and cornstarch in medium bowl until mixture is pale yellow and thick, about 1 minute; set aside. Heat milk, salt, and remaining 6 tablespoons sugar in medium saucepan over medium heat until simmering, stirring occasionally to dissolve sugar.
6. Gradually whisk half of milk mixture into yolk mixture to temper. Return milk-yolk mixture to saucepan. Return to simmer over medium heat and cook, whisking constantly, until mixture is thickened and 3 or 4 bubbles burst on surface, about 1 minute. Off heat, whisk in butter and vanilla. Transfer pastry cream to clean bowl, press parchment paper directly onto surface, and refrigerate until set, at least 3 hours.
7. FOR THE FROSTING: Using stand mixer fitted with whisk attachment, whip butter on medium-high speed until smooth, about 20 seconds. Add sugar and salt; reduce speed to medium-low and whip until most of sugar is moistened, about 45 seconds. Scrape down bowl, increase speed to medium, and whip until mixture is fully combined, about 15 seconds. Scrape down bowl, add cream and vanilla, and whip until incorporated, about 10 seconds. Increase speed to medium-high; whip until light and fluffy, about 4 minutes, scraping down bowl as needed. Add lemon zest and mix until combined.
8. Place 1 cake layer on cake turntable. Spread 1 cup pastry cream evenly over top. Repeat with remaining 2 cake layers and remaining 2 cups pastry cream. Spread frosting in even layer around sides of cake. While spinning cake turntable, run spatula horizontally from bottom to top of side of cake to create textured stripes.
9. Arrange blueberries in single row around base of cake. Alternate raspberries and blackberries around top edge of cake. Scatter remaining berries over top of cake. Sift sugar evenly over berries. Serve.