Preheat oven to 350 degrees F (175 degrees C). Spray a 9x13-inch baking pan with cooking spray.
Combine brownie mix, oil, eggs, and water in a large bowl. Stir with a wooden spoon until smooth, about 50 strokes. Pour into prepared baking pan and spread evenly.
Bake in the preheated oven until top is dry and edges have started to pull away from the sides of the pan, about 28 minutes. Remove from oven and let cool, about 30 minutes. Cut into small squares.
Whisk rum cream liqueur, milk, and chocolate pudding mix in a bowl until smooth.
Place half of the brownie squares in the bottom of a trifle bowl. Drizzle in half of the fudge topping and caramel topping. Spoon half of the rum cream pudding on top. Spread a quarter of the whipped topping over pudding. Repeat layers with remaining brownie squares, fudge topping, caramel topping, pudding, and whipped topping.
Cover with plastic wrap and refrigerate until set, at least 4 hours.