Combine the strawberries, raspberries, cane sugar, lemon juice, and cornstarch in a saucepan. Cook over medium heat for 8 to 10 minutes, stirring and crushing the berries, until thick and jammy. Refrigerate until cold.
Heat the oven to 350 degrees F and line a baking sheet with parchment paper. Mix the flour, brown sugar, peanut butter, coconut oil, and sea salt until chunky crumbs form.
Bake the peanut butter cookie crumble for 10 to 12 minutes, stirring once. Cool completely, then freeze the crumble.
Whisk the coconut milk, oat milk, cane sugar, cornstarch, vanilla, and sea salt in a saucepan. Cook over medium heat until steaming and slightly thickened, then whisk in the peanut butter until smooth.
Refrigerate the peanut butter base for at least 4 hours. Churn the cold base according to the ice cream maker's directions, adding two-thirds of the cookie crumble during the final 2 minutes.
Layer the ice cream with the berry swirl and remaining peanut butter cookie crumble. Freeze for 4 to 6 hours, then soften for 5 to 10 minutes before scooping.