Combine mixed berries, lemon juice, sugar, vanilla extract, and salt in a saucepan. Cook over medium heat for approximately 10 minutes, stirring and breaking up the berries, until thickened.
Transfer the berry mixture to a blender and blend until smooth, or use an immersion blender directly in the pot. Pass the sauce through a fine sieve to remove seeds. Cool the sauce and refrigerate.
In a food processor, blitz graham crackers, granulated sugar, and salt until finely ground. Stream in melted butter and pulse until the mixture is cohesive. Line a 23×23-centimetre baking pan with parchment paper. Evenly press the crust mixture into the bottom using the back of a measuring cup or fingers. Freeze while preparing the filling.
In a chilled mixing bowl, beat the cold heavy cream with a hand mixer until stiff peaks form, about 2–3 minutes. Start at a low speed and increase gradually. Take care not to over-beat. Refrigerate whipped cream until needed.
In a separate bowl, beat cream cheese and powdered sugar together until smooth. Add sour cream, vanilla extract, lemon juice, and salt. Continue mixing until thoroughly blended.
Gently fold the whipped cream into the cream cheese mixture using a rubber spatula, mixing just until incorporated.
Spread the cheesecake filling evenly over the prepared crust in the pan. Drop spoonfuls of berry sauce over the surface and use a knife to marble and swirl gently. Reserve leftover berry sauce for serving.
Cover the pan with plastic wrap and refrigerate for at least 8 hours or overnight, allowing the bars to fully set before slicing.