Toss 1 cup each strawberries and mixed berries with 1 tablespoon granulated sugar and the juice of 1/2 lemon in a medium bowl; set aside to macerate, 30 minutes.
Meanwhile, puree 1 cup mixed berries in a blender until smooth. Refrigerate the remaining 2 cups each strawberries and mixed berries for topping.
Beat the cream cheese, confectioners' sugar, vanilla, the zest of both lemons and the juice of 1 lemon in a large bowl with a mixer on medium-high speed until fluffy, about 4 minutes. Add the heavy cream and beat until slightly stiff peaks form, 1 to 2 minutes. Transfer half of the whipped cream mixture to a separate bowl and fold in the berry puree.
Line a 9-by-5-inch loaf pan with plastic wrap, leaving an overhang on all sides. Spread 1 cup of the plain whipped cream mixture in the pan (refrigerate the rest for topping). Cover with a single layer of cookies. Spread 1 cup of the berry whipped cream over the cookies, then top with half of the macerated berries and another 1 cup berry whipped cream. Repeat with another layer of cookies, berry whipped cream and the remaining macerated berries and berry whipped cream. Top with another layer of cookies (you may have some left over). Cover with plastic wrap and refrigerate at least 6 hours or overnight.
Before serving, toss the remaining 2 cups each strawberries and mixed berries with the remaining 1 tablespoon granulated sugar and juice of 1/2 lemon in a large bowl; set aside to macerate, 30 minutes.
Uncover the cake and invert onto a platter; remove the plastic. Cover with the reserved plain whipped cream. Top with the macerated berries.