Preheat the oven to 350 degrees F (175 degrees C). Spray a 9x13-inch baking pan with cooking spray.

Whisk flour, baking powder, and salt together in a medium bowl. Beat eggs and 1 cup sugar together on medium speed in a stand mixer until thick and light yellow, about 4 minutes. Add coconut and vanilla extracts. Turn mixer to low and add the flour in batches, alternating with the melted butter, and ending with the flour.

Pour batter into the prepared pan and spread into an even layer.

Bake in the preheated oven until golden brown and a cake tester comes out clean, 30 to 35 minutes. Let cool for 30 minutes.

Whisk 1 can coconut milk, sweetened condensed milk, and evaporated milk together in a medium bowl. Poke holes all over the cake with a wooden skewer and pour the milk mixture over the top. Let cake cool for at least 2 hours, or overnight. Place 2 remaining cans of coconut milk in the refrigerator to chill.

Scoop out the solid coconut cream from the tops of the cans, discarding the coconut water at the bottom. Whisk coconut cream and remaining 1/4 cup sugar together with an electric hand mixer until stiff peaks form. Spread over the top of the cake. Garnish with toasted coconut, lime, and sea salt. Cut into squares and serve.