Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x5-inch loaf pan.
Mix 1/4 cup light brown sugar, dark brown sugar, walnuts, 1 teaspoon cinnamon, and 1/4 teaspoon ginger together in a bowl. Cut in 1 1/2 teaspoon butter with a pastry blender until mixture is crumbly. Divide streusel mixture between two bowls. Mix chocolate chips into one bowl.
Combine 1/2 cup butter, 1/2 cup light brown sugar, and white sugar in a large bowl; beat with an electric mixer until light and fluffy. Add eggs one at a time, beating well after each addition. Add pumpkin puree, sour cream, dark rum, and vanilla extract; mix well.
Combine flour, 1 teaspoon cinnamon, baking soda, and sea salt in a bowl. Beat gradually into the pumpkin mixture until batter is smooth.
Spoon half of the batter into the prepared pan. Sprinkle chocolate chip streusel over batter, keeping it away from the sides of the pan. Spoon remaining batter on top, spreading it to the edges of the pan. Sprinkle remaining streusel on top.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 55 minutes. Transfer pan to a wire rack; let cool for 30 minutes. Invert cake onto the wire rack to cool completely.
Melt bittersweet chocolate in a microwave-safe glass or ceramic bowl in 15-second intervals, stirring after each melting, 1 to 3 minutes. Drizzle over the top and side of the cake. Chill cake until chocolate is set, about 20 minutes.