Preheat the oven to 375 degrees F (190 degrees C). Grease 12 muffin cups with 2 teaspoons coconut oil.
Soak raisins in warm water in a small bowl until rehydrated, about 10 minutes; drain. Chop raisins.
Beat 1/4 cup coconut oil, butter, and brown sugar together in a bowl. Add eggs, mashed bananas, applesauce, and grated carrot; beat to incorporate. Gently fold bread flour, whole-wheat flour, flax seed, chopped raisins, baking soda, salt, and vanilla extract into the banana mixture using a rubber spatula; pour into prepared muffin cups.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 18 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.