Whisk all-purpose flour, pastry flour, 1 tablespoon sugar and 3/4 teaspoon salt in a medium bowl. Crumble lard (or shortening) into the bowl and toss to coat. Add 3 tablespoons oil, 2 tablespoons water and vinegar; stir to evenly moisten the dough. Add more water, 1 tablespoon at a time, if necessary. Shape the dough into a 4-inch disk. Wrap in plastic and refrigerate for at least 1 hour or for up to 1 day.
Mix blackberries, blueberries, raspberries, strawberries, cornstarch and lime juice in a medium bowl with the remaining 2/3 cup sugar and 1/8 teaspoon salt. Let stand, stirring occasionally, for 15 minutes.
Position racks in middle and lower third of oven; preheat to 425 degrees F. Place a rimmed baking sheet on the lower rack (to catch any drips).
Roll the dough between two pieces of floured parchment into an 11-inch circle. Peel off the top piece of parchment and carefully invert the dough into a 9-inch pie pan (not deep-dish). Peel off the remaining parchment. Patch any rips in the dough. Trim and crimp the edges.
Fill the crust with the berry mixture. Mix oats, brown sugar and the remaining 3 tablespoons oil in a medium bowl. Sprinkle over the berries.
Bake the pie for 20 minutes. Reduce oven temperature to 350 degrees F. Continue baking until the filling is bubbling, 50 to 60 minutes more. Let cool completely before serving, about 1 1/4 hours.
To make ahead: Refrigerate dough (Step 1) for up to 2 days.
Equipment: Parchment paper, 9-inch pie pan (not deep-dish)