Preheat the oven to 350°F (175°C). Lightly spray an 8×8-inch baking pan with nonstick spray.
In a bowl, mix the crushed peanut butter sandwich cookies (reserve 8–9 cookies for topping) with the melted unsalted butter until all crumbs are evenly moistened.
Press the crumb mixture firmly into the bottom of the prepared pan to form an even crust.
Bake the crust for 10 minutes, then remove and let it cool completely in the pan.
In a small bowl, whisk or beat the instant chocolate pudding mix with 1½ cups cold milk until thickened according to package directions.
Spread the thickened chocolate pudding evenly over the cooled crust using a spatula, then refrigerate while you prepare the peanut butter layer.
In a medium bowl, beat the softened cream cheese on medium speed until smooth (about 1 minute).
Add the creamy peanut butter and beat on medium-high until smooth, about 30–60 seconds.
Reduce mixer speed to low and slowly add the powdered sugar, mixing until fully incorporated and the filling is smooth.
Remove the pan from the refrigerator and evenly spread the peanut butter mixture over the chocolate pudding layer.
Spread the 8-ounce tub of whipped topping evenly over the peanut butter layer.
Sprinkle the milk chocolate chips and peanut butter chips evenly over the whipped topping, then sprinkle the reserved crushed sandwich cookies on top.
Refrigerate the assembled bars for at least 2 hours to set before cutting into 2-inch x 2-inch squares.