Preheat your oven to 350°F (175°C). Grease an 8x8-inch or 9x9-inch square baking dish. In a large bowl, combine the flour, oats, brown sugar, cinnamon, and salt. Add the cold, cubed butter. Using a pastry cutter, two forks, or your fingertips, work the butter into the dry ingredients until the mixture resembles coarse crumbs with some pea-sized pieces remaining.
Reserve 1 cup of the crumble mixture for the topping. Press the remaining crumble firmly and evenly into the bottom of the prepared baking dish. Bake for 15-18 minutes, or until the edges are lightly golden. Remove from the oven and let it cool completely on a wire rack.
While the base cools, prepare the berries. In a medium saucepan, combine the strawberries, blueberries, 1/4 cup sugar, cornstarch, and lemon juice. Cook over medium heat, stirring frequently, for about 8-10 minutes, until the berries break down and the juices have thickened to a glossy, syrupy consistency. Remove from heat and let it cool to room temperature.
In a large bowl, beat the softened cream cheese, 1/3 cup sugar, and vanilla extract with an electric mixer on medium speed until smooth and creamy. In a separate, clean bowl, whip the cold heavy cream until stiff peaks form. Gently fold the whipped cream into the cream cheese mixture until no white streaks remain.
Spread the cooled cream cheese mixture evenly over the completely cooled crumble base. Carefully spoon the cooled berry compote over the cream layer, spreading it gently to cover. Sprinkle the reserved 1 cup of crumble topping evenly over the berries.
Bake the assembled dessert for 20-25 minutes, or until the crumble topping is golden brown and the edges are bubbly. Remove from the oven and let it cool on a wire rack for at least 1 hour. Then, refrigerate for a minimum of 3-4 hours, or preferably overnight, to fully set.
To serve, use a sharp knife to cut into 9 clean squares. Wipe the knife between cuts for neat edges. Plate the squares and garnish with additional fresh strawberries and blueberries. Dust with powdered sugar if desired.