Line an 8" square baking pan with parchment paper or aluminum foil, leaving an overhang on two opposite sides for easy removal; set the pan aside.
Preheat the oven to 375°F.
Make the crust: in a mixing bowl, combine ¼ cup melted unsalted butter (cooled completely) and ¼ cup granulated sugar; stir until blended.
Add ¾ cup all-purpose flour, ¼ teaspoon baking soda, ¼ teaspoon baking powder, and ¼ teaspoon salt to the butter mixture and mix until a thick dough forms.
Press the dough evenly into the bottom of the prepared pan to form the crust; set the pan aside.
Make the filling: in a clean mixing bowl, beat 8 oz softened cream cheese with ½ cup granulated sugar until smooth and free of lumps.
Add 2 tablespoons all-purpose flour, 1 large egg, and ½ teaspoon pure vanilla extract to the cream cheese mixture; mix until just combined and smooth, scraping the bowl as needed.
Stir 1.5 cups sweetened shredded coconut into the filling until evenly distributed.
Pour the coconut cheesecake mixture over the prepared crust and spread it into an even layer.
Bake in the preheated oven for 33 to 36 minutes, until the top is golden and the center no longer jiggles when the pan is gently shaken.
Cool the bars in the pan on a wire rack until completely cool. Use the parchment or foil overhang to lift the bars from the pan, then slice.