Place the blackberries in a blender and blend until smooth. Pour purée through a mesh sieve and into a saucepan to remove the seeds. Discard seeds and solids.
Add sugar to blackberry purée in the saucepan. Bring to a simmer and cooker over medium low heat until sugar dissolves, about 5 minutes, stirring occasionally.
Stir together cornstarch and 1 tablespoon of water. Add to the mixture in the saucepan. Stir frequently until thickened, 3 to 5 minutes.
Continue cooking over low to medium-low heat, stirring occasionally and scraping the sides of the pot, for 5 to 10 minutes, or until mixture is thickened to a jelly/jammy consistency. You should be able to drag a spatula across the bottom of the pot and have the mixture very, very slowly spread back into place.
Remove from heat and let the reduction cool completely. Cover with plastic wrap and store in the refrigerator until thoroughly chilled, or overnight. (Reduction will thicken more as it cools.)
Reduction can be made 1-2 days in advance and stored covered in the refrigerator.
To make the cake:
Preheat oven to 350°F. Butter and flour three 8-inch cake pans.
Whisk together flour, baking powder, baking soda, salt, lime zest, and sugar.
Add sugar, eggs, oil, coconut extract, and milk. Mix on medium speed with an electric mixer until all ingredients are combined. Fold in shredded coconut.
Divide batter evenly among cake pans. Bake for 23-25 minutes or until cakes are golden brown and set in the center. (Test by inserting a toothpick into the center of each cake layer. If it comes out clean, they're done!)
Let cakes cool in pans for about 10 minutes before transferring to a wire rack to cool completely.
To make the frosting:
Cream butter on medium-high speed with an electric mixer. With mixer running, add shortening. Mix until smooth and creamy.
Add blackberry reduction, coconut extract, and cream; mix well.
Add half of the powdered sugar and mix until well blended. Repeat with the second half of powdered sugar, and mix for 1 minute, scraping the sides of the bowl as necessary.
To make the lime sugar:
To make the lime sugar, place zest and coarse sugar in a bowl. Use your fingers to rub the zest and sugar together until fragrant and the sugar has a light greenish hue.
Assembly:
When the cake layers are cool, level them with a cake saw or serrated knife. Place one cake layer on a cake stand or plate. Pipe a border of frosting around the top, then fill with half of the remaining black berry reduction.
Place the second cake layer on top and repeat with the border of frosting and the rest of the blackberry reduction.
Spread remaining frosting around the outside and top of the cake. Garnish with fresh blackberries, lime slices, toasted coconut, and lime sugar.