Sift flour, 2 1/4 cup sugar, baking powder, and salt together in a bowl; cut shortening into flour mixture until fine crumbs are formed.
Beat egg whites, milk, strawberry syrup, and vanilla extract into flour mixture with an electric mixer on low for 1 minute; increase to high until smooth and blended, about 2 more minutes, scraping the sides of the bowl frequently.
Pour batter into prepared baking pans.
Bake in the preheated oven until a toothpick inserted in the center of each cake comes out clean, 25 to 30 minutes.
Place chocolate in a small microwave-safe bowl; heat on High in microwave until melted, about 1 minute. Stir chocolate until smooth.
Beat cream cheese and 1/3 cup sugar in a large bowl with an electric mixer on High until combined, 1 minute; stir chocolate into mixture until well mixed.
Gently fold whipped topping, pudding mix, and chocolate sandwich cookies into cream cheese mixture to form the filling.
Spread about 1/3 filling on top of one cake. Place second cake atop filling layer. Repeat with a filling layer, the third cake, and a final layer of filling on top.