In a medium saucepan, whisk together the sugar, cornstarch, and salt, then whisk in the coconut milk and corn syrup. Cook over medium heat, whisking constantly, for 5 to 7 minutes, until the base simmers and thickens slightly.
Remove from the heat and stir in the vanilla. Pour into a bowl, press plastic wrap onto the surface, and refrigerate for at least 6 hours or overnight.
In a medium skillet, toss the peaches with the brown sugar, cinnamon, and cornstarch, then add the lemon juice and butter. Cook over medium heat, stirring often, for 10 to 12 minutes, until the peaches are tender and the juices turn thick and syrupy, then cool and refrigerate until cold.
Preheat the oven to 350°F and line a baking sheet with parchment paper. Stir together the flour, sugar, baking powder, cinnamon, and salt, then drizzle in the melted butter and mix until large, buttery clumps form.Scatter the clumps over the baking sheet and bake for 12 to 15 minutes, until golden. Let cool completely, then break into bite-size pieces.
Pour the chilled base into an ice cream maker bowl that has been frozen for at least 24 hours. Churn for 20 to 25 minutes, until thick and creamy like soft serve.
Spread one third of the ice cream into a 9x5-inch loaf pan, then top with spoonfuls of cinnamon peaches and a handful of cobbler crumble. Repeat the layers two more times, then gently swirl with a skewer.
Cover tightly and freeze for 4 to 6 hours, until firm. Let sit at room temperature for 10 minutes before scooping.