Preheat oven to 350°F (177°C). Grease a 9-inch springform pan and line the bottom with parchment paper.
Beat softened butter and ¾ cup sugar until light and fluffy, about 2-3 minutes. Add eggs one at a time, then mix in vanilla and sour cream.
Whisk together flour, baking powder, baking soda, and salt. Gradually add dry ingredients to wet mixture, mixing just until combined.
Toss blackberries with 1 tablespoon flour, then gently fold them into the batter. Spread evenly into the prepared pan.
Beat cream cheese and ¼ cup sugar until smooth. Add egg and vanilla, mixing until no lumps remain.
Spoon dollops of cream cheese mixture over the batter and gently spread into an even layer using an offset spatula.
Whisk flour, brown sugar, cinnamon, and salt. Cut in cold cubed butter until the mixture resembles coarse, pea-sized crumbs.
Sprinkle crumb topping evenly over the cream cheese layer. Scatter extra blackberries on top for visual appeal.
Bake for 55-65 minutes. Tent with foil at the 40-minute mark if browning too quickly. Cake is done when center no longer jiggles.
Cool in the pan for 30 minutes before releasing the springform sides. If using glaze, whisk powdered sugar and lemon juice and drizzle over the warm cake.