Preheat oven to 350°F (175°C). Line a muffin tin with paper cupcake liners.
In a large bowl, whisk together the 3 2/3 cups flour, 1 tablespoon baking soda, 1 teaspoon salt, and 1 cup bran until evenly combined.
In a separate bowl, combine the melted-and-cooled 1 1/2 cup butter with the 3/4 cup light brown sugar, 2/3 cup molasses, 1 cup shredded zucchini, and 3 cups sour cream. Stir until smooth and uniformly mixed.
Pour the wet mixture into the dry ingredients. Stir gently with a spatula or wooden spoon just until the batter is combined—do not overmix.
Carefully fold in the 1 1/2 cups blackberries with a spatula, folding just until the berries are distributed and minimizing pressure so they do not burst.
Spoon the batter into the prepared liners, filling each about two-thirds to three-quarters full.
Bake at 350°F for about 30 minutes, or until a toothpick inserted into the center of a muffin comes out clean or with a few moist crumbs.
Let the muffins cool in the pan for 5–10 minutes, then transfer them to a wire rack to cool completely.