Preheat the oven to 350°F (175°C). Line the bottom of an 18x13-inch rimmed baking sheet with parchment paper; do not grease.
In the bowl of a stand mixer fitted with the whisk, add 5 room-temperature eggs and beat on high for 1 minute until foamy.
With the mixer running, gradually add 3/4 cup granulated sugar and continue beating on high for 7 minutes, until the mixture is thick, fluffy, and has roughly tripled in volume.
Whisk together 3/4 cup all-purpose flour and 1/2 tsp baking powder. Sift the flour mixture into the egg batter in three additions, folding gently after each addition until no streaks of flour remain.
Spread the batter evenly in the prepared baking sheet and bake at 350°F for 13–15 minutes, until the top is golden and the cake springs back lightly.
Immediately after removing from the oven, run a thin spatula around the edges to loosen the cake. Invert the hot cake onto a clean, dry linen towel, peel off the parchment, and roll the cake up in the towel. Allow to cool to room temperature, about 30–60 minutes, in a draft-free place.
While the cake cools, make the lemon syrup by combining 1/4 cup warm water, 1 Tbsp granulated sugar, and 4 Tbsp fresh lemon juice; stir until the sugar dissolves and set aside.
For the filling, beat 8 oz softened cream cheese, 8 Tbsp softened unsalted butter, and 1 cup powdered sugar together, starting on low to incorporate then increasing to high for 2–3 minutes until white and fluffy, scraping the bowl as needed.
Fold in the coarsely chopped blackberries (reserve 6 whole berries for garnish) with the mixer on low until just combined.
Carefully unroll the cooled cake and brush the surface evenly with the lemon syrup. Dot the blackberry cream filling over the cake and spread into an even layer.
Re-roll the cake tightly in the same direction as before. Trim the ends if desired and dust the outside generously with powdered sugar.
Beat 3/4 cup chilled heavy whipping cream with 1 Tbsp powdered sugar on high for about 2 minutes until fluffy. Transfer to a piping bag fitted with a large open star tip and pipe six rosettes on top of the roll.
Place a reserved whole blackberry and a lemon triangle in the center of each piped rosette to garnish, then serve.