Preheat oven to 350°. Line bottom and sides of an 8-inch square pan with parchment paper, allowing 3 inches to extend over sides. Beat softened butter and 1/4 cup sugar at medium speed with an electric mixer until smooth. Stir together 1 cup all-purpose flour and 1/4 tsp. table salt; gradually add to butter mixture, beating at low speed just until blended after each addition. Press dough into bottom of prepared pan. Bake 20 to 25 minutes or until lightly browned. Whisk together 1 1/2 cups sugar, 1/2 cup all-purpose flour, Key lime zest, and 1/4 tsp. salt in a large bowl. Whisk in 1 egg and 3 egg whites just until blended. Process blackberries in a blender until smooth. Pour pureed berries through a fine wire-mesh strainer into sugar mixture, discarding seeds; whisk in Key lime juice. Pour over warm crust; bake 30 to 35 minutes or until center is set. Cool on wire rack 30 minutes. Cover with plastic wrap; chill 2 hours. Lift from pan, using parchment paper as handles. Cut into bars.