Make the crust: Pulse the flour, granulated sugar and salt in a food processor to combine. Add the shortening and pulse a few times until coarse crumbs form. Pulse in the butter until just combined with some pea-size pieces remaining. Pulse in the vinegar and 2 tablespoons ice water until the dough starts to come together but is still crumbly, adding up to 2 more tablespoons ice water if needed. Turn out the dough onto a piece of plastic wrap and pat into a disk. Wrap and refrigerate until firm, at least 1 hour or overnight.
Meanwhile, make the topping: Combine the oats, flour, brown sugar, salt, cinnamon and allspice in a medium bowl. Work in the butter with your fingers until clumps form. Refrigerate until ready to assemble the pie.
Make the filling: Combine the berries, granulated sugar, brown sugar, lemon juice, tapioca, cinnamon, cardamom and salt in a large bowl; toss well.
Place a baking sheet on the lowest oven rack and preheat to 425 degrees F. Roll out the dough into a 12-inch round on a lightly floured surface; ease into a 9-inch pie plate and crimp the edge. Refrigerate until firm, at least 20 minutes.
Pour the berry filling into the crust and sprinkle evenly with the crumb topping. Transfer the pie to the hot baking sheet and bake 20 minutes. Reduce the oven temperature to 375 degrees F and continue baking, rotating the pie halfway through, until the crust and topping are browned and the filling is bubbling, 35 to 40 more minutes. (Tent loosely with foil if the pie is browning too quickly.) Transfer to a rack and let cool completely.