Make the cake: Preheat the oven to 350 degrees F. Generously butter a nonstick 12-cup Bundt pan. Whisk 3 cups flour, the baking powder and salt in a medium bowl.
Beat 2 sticks butter, the granulated sugar and vegetable oil in a bowl with a mixer on medium-high speed until fluffy, at least 5 minutes, scraping down the sides of the bowl with a rubber spatula as needed. Reduce the mixer speed to low; beat in the eggs one at a time, then beat in the vanilla. Add about one-third of the flour mixture and half of the buttermilk; beat until almost incorporated. Add another one-third of the flour mixture and the remaining buttermilk. Beat, scraping down the sides of the bowl as needed, until just combined. Add the remaining flour mixture and beat 30 seconds. Finish incorporating the flour by hand to avoid overmixing.
Toss the blueberries with the remaining 2 tablespoons flour in a small bowl. Spoon one-third of the batter evenly into the prepared pan. Sprinkle in half of the blueberries, then top with another one-third of the batter. Scatter the remaining blueberries on top and cover with the rest of the batter; smooth the top. Bake until the cake is golden brown and a toothpick inserted into the center comes out clean, 1 hour to 1 hour, 10 minutes. Transfer to a rack and let cool 30 minutes in the pan. Run a small sharp knife around the edge of the pan to loosen the cake, then invert onto the rack to cool completely.
Meanwhile, make the toppings: Toss the strawberries with the granulated sugar in a bowl; set aside to macerate, 30 minutes. Just before serving, make the glaze: Whisk the confectioners' sugar, butter and 4 tablespoons milk in a bowl; if the glaze is too thick, whisk in up to 1 more tablespoon milk, a little at a time. Pour the glaze over the cake, letting it drip down the sides. Serve with the strawberries and their juices.