Prep the pan. Heat the oven to 325°F (165°C).
Line the bottom of a springform pan with parchment. Wrap the outside with foil in case of leaks. Lightly mist the inside with nonstick spray.
Make the crust. In a bowl, stir crushed crackers, sweetener, salt, and melted light butter until evenly moistened.
Press firmly into the bottom of the pan with the base of a measuring cup for a compact layer. Bake for 8–10 minutes, then set aside to cool slightly.
Cook the blackberry swirl. In a small saucepan, combine blackberries, sweetener, and lemon juice. Cook over medium heat, stirring and gently mashing, for 4–5 minutes until juicy.
Stir in the cornstarch slurry and simmer 30–60 seconds until slightly thick. Take off heat. For a smooth texture, push through a strainer to remove seeds.
Cool to room temperature.
Mix the filling. In a large bowl, beat light cream cheese with sweetener on medium speed until smooth and fluffy, about 2 minutes. Add Greek yogurt, lemon zest, lemon juice, vanilla, cornstarch, and salt. Mix on low until combined.
Add eggs one at a time, mixing on low just until incorporated. Do not overmix—too much air causes cracks.
Assemble. Pour the filling over the crust and smooth the top. Drizzle spoonfuls of blackberry sauce over the surface. Use a skewer or butter knife to gently swirl.
Reserve a little sauce for serving if you like.
Bake gently. Place the pan on a baking sheet. Bake at 325°F (165°C) for 40–55 minutes, depending on pan size. The edges should be set and the center should still jiggle slightly like gelatin. Do not bake until completely firm.
Cool slowly. Turn off the oven, crack the door, and let the cheesecake rest inside 30 minutes.
Move to a rack and cool fully to room temperature.
Chill. Cover and refrigerate at least 6 hours or overnight. This helps it set and deepens the lemon-blackberry flavor.
Serve. Run a thin knife around the edge before releasing the springform. Slice with a warm, clean knife.
Garnish with fresh blackberries, a light dollop of Greek yogurt or whipped topping, and an extra drizzle of sauce if desired.