Apple-Cinnamon Bread Pudding with Ginger Ale Sauce

3 eggs, beaten
2 cups milk, cream, or combination
1 1/2 teaspoons rum extract
1/2 teaspoon salt
1/4 cup brown sugar
6 cups cinnamon bread, cubed (about 15 slices)
1 (20 1/2-ounce) can apple pie filling
1/2 cup raisins
2 packets apple cinnamon instant oatmeal (recommended: Quaker Oats)
Sauce:
1/2 cup ginger ale
1 package whipped topping mix (recommended: Dream Whip)
1/2 teaspoon ground cinnamon
1/2 teaspoon rum extract
1/4 teaspoon salt