Preheat oven to 300 degrees F (150 degrees C). Spray a 9x13-inch baking dish with cooking spray.
In a bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, white sugar, and brown sugar until thoroughly combined. In a large bowl, beat the eggs and milk together until smooth, and add the flour mixture, alternating with the persimmon pulp in several additions, mixing well after each addition. Stir in the melted butter. Scrape the batter into the prepared baking dish.
Bake in the preheated oven until a toothpick inserted into the pudding comes out clean, about 1 hour. Allow to cool before serving.