Preheat oven to 350°F. Line length of 13-by-9-inch cake pan with parchment paper, leaving a 4-inch overhang. Lay another sheet across width of pan and up sides (without overhang). Spray paper evenly with cooking spray.
Make cake: Whisk together flour, granulated sugar, cocoa powder, baking soda, baking powder and salt in a large bowl. Using an electric mixer, beat in butter on low speed until mixture is crumbly. Add buttermilk, eggs and vanilla. Increase speed to medium and beat, scraping down inside of bowl, until smooth, 2 to 3 minutes. Transfer to baking pan.
Bake cake until a cake tester inserted in center comes out clean, about 30 minutes. Cool in pan on a rack for 10 minutes. Holding overhanging pieces of paper, pull cake from pan and let cool completely on rack. Invert platter on cake and flip over to invert cake. Carefully peel off paper and discard.
Make frosting: Using an electric mixer, beat cream cheese and butter on medium speed until light. Gradually beat in confectioners' sugar until fluffy. Beat in lemon juice. Refrigerate frosting for about 10 minutes if it's too soft to spread.
Frost sides and top of cake. Arrange blueberries in a rectangle in upper left-hand corner of cake. Arrange raspberries in 7 rows for stripes. Refrigerate cake until serving. (Cake may be made up to 3 hours ahead.)