Combine mixed berries, lemon juice, sugar, vanilla extract, and salt in a saucepan and cook over medium heat for about 10 minutes until thickened, stirring frequently and mashing berries.
Puree the berry mixture in a blender or with an immersion blender, then strain through a fine sieve to remove seeds. Cool and refrigerate.
In a food processor, grind graham crackers, granulated sugar, and salt until finely ground, then add melted butter and pulse until mixture is cohesive.
Line a 23x23 cm baking pan with parchment paper and firmly press the crust mixture evenly into the bottom. Freeze crust while preparing filling.
Beat cold heavy cream with a hand mixer until stiff peaks form, then refrigerate.
In a separate bowl, beat cream cheese and powdered sugar until smooth. Add sour cream, vanilla extract, lemon juice, and salt, mixing well.
Gently fold whipped cream into cream cheese mixture until just combined.
Spread cheesecake filling evenly over crust. Drop spoonfuls of berry sauce on top and gently marble using a knife. Reserve extra sauce for serving.
Cover with plastic wrap and refrigerate for at least 8 hours or overnight to set before slicing and serving.