Preheat oven to 350 degrees F (175 degrees C). Spray the bottom of a 10-inch springform pan with cooking spray.
Measure 1 cup cake mix; set aside for crumb topping. Place the remaining cake mix in a large bowl; add 3/4 cup plus 1 tablespoon milk, eggs, and oil. Beat cake mix mixture using an electric mixer until batter is thoroughly combined, about 2 minutes. Pour batter into the prepared pan.
Mix melted butter and vanilla extract together in a bowl; stir in reserved 1 cup cake mix until mixture is crumbly. Sprinkle crumbs over top of cake batter.
Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, 30 to 35 minutes. Cool cake to room temperature in the pan.
Beat cream cheese, 1/3 cup confectioners’ sugar, lemon juice, and lemon zest together in a bowl until smooth and creamy. Beat cream and remaining 1/3 cup confectioners’ sugar together in a separate bowl using an electric mixer until stiff peaks form. Fold cream cheese mixture into whipped cream.
Remove cake from springform pan. Cut cake horizontally into 2 layers using a serrated knife; remove top layer. Spread filling onto the bottom cake layer; place top cake over filling. Refrigerate cake for at least 4 hours. Dust cake with more confectioners’ sugar before serving.