Preheat oven to 375°F (190°C). Unroll one pie crust onto a lightly floured surface. Roll slightly to make it 1 inch wider than your tart pan. Transfer to tart pan, pressing into bottom and up sides. Trim overhang. Prick bottom with fork. Line with parchment and fill with pie weights or dried beans.
Bake crust with weights for 12 minutes. Remove parchment and weights. Bake for another 2-3 minutes until lightly golden and dry. Set aside to cool completely.
In a food processor or blender, combine 12 oz blackberries (2 containers), 1/4 cup confectioners' sugar, and 2 tablespoons lime juice. Blend until smooth. Strain through a fine-mesh sieve into a bowl, pressing with a spatula. Discard seeds. Set jam swirl aside.
In a large bowl using an electric mixer, beat room-temperature mascarpone on medium speed until smooth and creamy, about 1 minute. Scrape down the bowl.
Add 1/2 cup confectioners' sugar, 1 tablespoon lime juice, and 1 1/2 tablespoons chopped fresh mint to the mascarpone. Beat on low until combined, then increase to medium for 30 seconds until light and fluffy. Do not over-beat.
Spread mascarpone filling evenly into cooled tart crust using an offset spatula or the back of a spoon.
Drop spoonfuls of blackberry-lime jam swirl over the filling (about 8-10 dollops). Use a knife or toothpick to gently swirl the jam into the mascarpone, creating ribbons. Do not over-mix.
Refrigerate tart for at least 2 hours, or up to 24 hours, until filling is set.
Just before serving, arrange remaining 6 oz fresh blackberries on top of the tart in a decorative pattern. Sprinkle with lime zest and fresh mint leaves. Dust lightly with additional confectioners' sugar through a fine-mesh sieve, if desired.
Slice with a sharp knife, wiping the blade clean between cuts for neat slices. Serve chilled or at cool room temperature.