Preheat your oven to 350°F (175°C). Line a 15x10-inch jelly roll pan with parchment paper, allowing some overhang on the sides for easy removal later.
In a large mixing bowl, beat the 5 large eggs and 3/4 cup of granulated sugar with an electric mixer on high speed until the mixture is thick and pale, about 5 minutes.
Gently fold in the 3/4 cup of all-purpose flour and 1/2 tsp of baking powder using a spatula. Be careful not to overmix; you want to keep the batter light and airy.
Stir in the 1/4 cup of warm water and 4 Tbsp of fresh lemon juice until just combined.
Pour the batter into the prepared jelly roll pan and spread it evenly. Bake in the preheated oven for 12-15 minutes, or until the cake springs back when touched.
Once baked, remove the cake from the oven and immediately dust it with 1 Tbsp of granulated sugar. Carefully lift the cake using the parchment paper and roll it up from the short end. Let it cool completely while rolled, to maintain its shape.
In a medium bowl, beat together the 8 oz softened cream cheese and 8 Tbsp softened unsalted butter until smooth. Gradually add in the 1 cup of powdered sugar until well combined.
Gently fold in the chopped blackberries into the cream cheese mixture until evenly distributed.
In a separate bowl, whip the chilled 3/4 cup of heavy whipping cream with 1 Tbsp of powdered sugar until soft peaks form.
Fold the whipped cream into the blackberry cream cheese mixture until well combined and fluffy.
Carefully unroll the cooled cake and spread the blackberry cream filling evenly over the surface. Roll it back up tightly, using the parchment paper to help guide it.
Wrap the cake roll in plastic wrap and refrigerate for at least 1 hour before serving. When ready, slice into pieces and garnish with reserved whole blackberries and lemon triangles.