Prepare the cake according to the package instructions and bake it in a 9 x 13-inch dish. Once baked, remove the cake from the oven and, using the handle of a wooden spoon, poke holes about 1 inch apart. Set the cake aside to cool.
In two small bowls, dissolve 2 teaspoons of gelatin in 2 tablespoons of water each and set aside to bloom.
To make the red sauce, combine about 1 cup of strawberries, 1 cup of raspberries, sugar, lemon juice, and 1 tablespoon of water in a medium saucepan.
Bring to a simmer and cook, stirring frequently, until the fruit softens, about 5 to 8 minutes. Using a potato masher, mash the berries, then add the dissolved gelatin along with the remaining diced strawberries and raspberries.
Stir over medium heat until the gelatin is completely melted, then set the sauce aside to cool.
For the blueberry sauce, combine about 1½ cups of blueberries, sugar, lemon juice, and 1 tablespoon of water in a medium saucepan. Bring to a simmer and cook until the berries soften, stirring frequently.
Mash the berries with a potato masher, add the dissolved gelatin and remaining blueberries, and stir over medium heat until the gelatin melts completely. Set aside to cool.
When both sauces have cooled and slightly thickened, spoon them alternately into the holes on the cake, filling each hole with red or blue sauce.
Use the back of the spoon to press as much sauce as possible into the holes, spreading any remaining sauce evenly over the top of the cake.
Refrigerate the cake to allow the sauce to set completely.
To prepare the topping, beat the softened cream cheese with ½ cup powdered sugar until smooth and set aside. In a separate bowl, whip the heavy cream with vanilla and ½ cup powdered sugar until stiff peaks form, yielding about 4 cups of whipped cream.
Divide the whipped cream in half.
Fold 2 cups of the whipped cream into the cream cheese mixture, scraping down the sides with a spatula to combine thoroughly, then spread this mixture evenly over the cake.
Transfer the remaining 2 cups of whipped cream into a piping bag fitted with a star tip and pipe decorative swirls on top of the cake, or simply spread it with a spoon over the cream cheese layer. Store the cake in the refrigerator until ready to serve.