Preheat the oven to 350 degrees F (175 degrees C).

Combine cake mix, water, 3 eggs, and oil in a large bowl; beat with an electric mixer on low speed until moistened. Increase speed to medium and beat for 2 minutes. Pour batter into a 2 1/2-quart oven-proof bowl; scrape sides of the bowl and set aside.

Beat cream cheese, sugar, and remaining 1 egg in a mixing bowl until combined; spoon into center of cake batter.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 1 hour 5 minutes.

Cool cake in the bowl for 10 minutes. Loosen cake from the bowl using a knife; invert onto a wire rack and let cool completely.

Meanwhile, whisk milk, pudding mix, and confectioners' sugar together in a medium bowl for 2 minutes. Stir in whipped topping. Refrigerate until ready to use.

Place cake on a serving plate. Frost with chilled pudding mixture; cover with coconut. Keep cake refrigerated.