Preheat the oven to 400F and place a rimmed baking sheet, to catch any juices, on the oven rack.
Place 3 ½ cups berries in a large saucepan and set over medium heat. Use a potato masher to mash and break up the berries.
Cook the mixture, stirring occasionally until the mixture is thickened and reduced to 1 ½ - 1 ¾ cups. Let cool slightly while preparing the remainder of the filling.
In a large bowl, combine both sugars, tapioca flour, salt, lemon juice, and remaining 3 ½ cups of berries.
Peel and grate the apple on a box grater. Place the apple in a clean kitchen towel and wring out as much liquid as you can. Place the grated apple in the bowl with the uncooked berry/sugar mixture and mix to combine.
Add the cooked, and slightly cooled berries to the mixture and stir until well combined.Pour the berry mixture into the chilled pie dough. Top the pie with the chilled crumble topping.
Place pie on the baking sheet and bake 30 minutes. Reduce temperature to 350F and continue to bake until the juices bubble, 30-35 minutes.
Cool to room temperature, about 3-4 hours, before serving.