Process cookies from 1 package in food processor until finely crushed. Place remaining cookies in a large zip-top freezer bag; coarsely crush with a rolling pin. Set aside.
Beat cream cheese and butter at medium speed with an electric mixer until creamy. Reduce speed to low; gradually add powdered sugar, beating until smooth.
In a separate bowl, whisk together milk and pudding mix for 2 minutes. Stir pudding mixture into cream cheese mixture. Stir in coarsely crushed cookies and toffee bits. Fold in half of whipped topping; fold in remaining topping. Cover and chill.
Spread one-third of finely crushed cookie crumbs in a 4-quart trifle dish. Layer one-third of strawberries, stemmed side down, over crumbs. Carefully spoon half of pudding mixture over strawberries. Repeat layers with one-third of crumbs, one-third of strawberries, and remaining half of pudding mixture. Top with remaining crumbs and strawberries. Cover and chill at least 3 hours. Store in refrigerator.