Preheat oven to 350°F. Process cookies in a food processor until finely ground, about 1 minute. Drizzle melted butter over cookie crumbs; pulse until well combined, 3 to 4 times. Press crumb mixture onto bottom of a lightly greased 13- x 9-inch baking dish. Bake in preheated oven until lightly browned, about 10 minutes. Cool completely, about 30 minutes. Reduce oven temperature to 325°F.

Meanwhile, combine raspberries and 1 tablespoon each of the sugar and water in a small saucepan over medium. Bring to a boil, and cook, stirring frequently to mash raspberries, until thickened, 6 to 8 minutes. Place raspberry mixture in a fine wire-mesh strainer over a bowl; press mixture until there are about 3 tablespoons raspberry puree in bowl. Discard solids in strainer. Cool puree completely. Repeat procedure with blueberries, 1 tablespoon sugar, and remaining 1 tablespoon water.

Beat softened cream cheese, salt, and remaining 3/4 cup sugar with an electric mixer on medium-low speed until smooth, 1 to 2 minutes. Add egg whites and egg, 1 at a time, beating well after each addition. Add sour cream, all-purpose flour, fresh lemon juice, and vanilla extract, and beat at low speed until smooth, 1 to 2 minutes. Pour batter into prepared baking dish; smooth top.

Drop level teaspoonfuls of raspberry and blueberry purees all over top of batter; gently swirl with a knife.

Bake at 325°F until cheesecake is almost set (center will still jiggle slightly when dish is touched), 25 to 28 minutes. Run a knife or offset spatula around edge of dish to loosen sides of cheesecake. Cool on wire rack at room temperature 1 hour. Cover and chill 4 hours or overnight. Cut into bars.