Preheat the oven to 350°F (175°C) and grease and flour three 8-inch round cake pans.
Combine the cake mix, 1 cup water, eggs, vegetable oil, and 1 teaspoon coconut extract in a bowl; beat with an electric mixer on medium for 2 minutes.
Divide batter evenly into pans and bake until lightly browned and a toothpick comes out clean, about 25 minutes.
Cool in pans 10 minutes, loosen edges with a knife, then invert onto wire racks and cool at least 30 minutes.
In a large bowl, beat the butter until creamy, then mix in the remaining 2 1/2 teaspoons coconut extract.
Gradually beat in 1 cup confectioners' sugar, then 1 tablespoon heavy cream.
Continue adding sugar, 1 cup at a time, alternating with cream, until smooth and creamy.
Level cake tops with a serrated knife.
Place one layer on a platter, frost with about 1/4 of the frosting, and sprinkle with 1/3 cup flaked coconut.
Place the final layer on top, frost top and sides, and press 1/3 cup coconut onto the sides.
To tint the remaining 1 cup coconut, place it in a resealable bag with 1/2 teaspoon water and green food coloring; shake and knead until evenly colored, then spread on a paper towel to dry.
Scatter the green coconut over the cake and nestle jelly beans in the center.