For the vanilla cake: Preheat the oven to 350 degrees F. Line three 9-inch round cake pans with parchment circles and spray lightly with cooking spray.
Combine the butter, oil and 2 cups sugar in the bowl of a stand mixer fitted with the paddle attachment and beat until light and fluffy, about 2 minutes. Beat in the eggs one at a time, scraping down the bowl between each egg. Beat in 1 tablespoon vanilla bean paste until incorporated.
Whisk together the flour, baking powder and salt in a medium bowl. Add a third of the dry ingredients to the mixer and beat on low speed. Add half of the buttermilk and beat until just incorporated. Alternate dry and wet ingredients, ending with the dry and scraping down the bowl as needed until the batter is smooth.
Divide the batter evenly among the prepared cake pans and bake until the cakes are lightly golden and the centers spring back to the touch, about 30 minutes. Let the cakes cool in their pans for 10 minutes, then turn them out onto a cooling rack to cool completely.
Meanwhile, to make the cake syrup, combine the remaining 1 cup sugar and 1 tablespoon vanilla bean paste with 1 cup water in a small saucepan and bring to a simmer over medium heat. Cook, stirring occasionally, until the sugar has dissolved, about 5 minutes. Remove from the heat and let cool to room temperature.
For the vanilla Swiss buttercream: Bring a medium saucepan filled with 2 inches of water to a simmer. Whisk together the egg whites and sugar in the bowl of a stand mixer and set the bowl over the simmering pot. Continue whisking until the sugar has dissolved and the mixture feels hot to the touch, 1 to 2 minutes. Wipe the condensation from the bottom of the bowl and attach it to the stand mixer fitted with the whisk attachment. Beat the egg white mixture on medium-high speed until the whites are glossy and the underside of the bowl is no longer warm, about 5 minutes. With the mixer running, begin adding a few chunks of butter at a time. Continue to add all the butter in this way until the buttercream is smooth and fluffy, about 2 minutes. Beat in the vanilla bean paste and salt.
For the assembly: Trim the tops of the cakes, if necessary. Using a pastry brush, brush the tops of each layer of cake with the cake syrup, allowing it to soak in slightly. Add about 1 1/2 cups of buttercream to the top of the first cake and smooth it into an even layer with an offset spatula. Place the second cake on top of the first and add another layer of buttercream. Place the last cake on top and frost just the top with the remaining buttercream. Arrange some of the blackberries, blueberries, raspberries and strawberries decoratively around the top edge of the cake. Serve with the remaining berries on the side.