Preheat oven to 350°F (175°C) and lightly coat a 9x13-inch baking dish with cooking spray. Combine softened cream cheese, Russian dressing, mayonnaise, and Dijon mustard in a bowl until smooth. Toss cubed rye bread with melted butter, garlic powder, and caraway seeds (if using). Spread half of the bread cube mixture across the bottom of the baking dish, pressing lightly. Distribute half of the chopped corned beef evenly over the bread layer. Spread half of the well-drained sauerkraut over the corned beef layer. Drizzle half of the cream cheese dressing mixture over the sauerkraut layer. Sprinkle half of the shredded Swiss cheese over the dressing layer. Repeat layers with remaining ingredients in the same order. Cover with aluminum foil and bake for 25 minutes. Remove foil and continue baking for 15 minutes until cheese is golden brown. Let stand for 5-10 minutes before serving. Garnish with sliced green onions.