In a small saucepan melt 1/2 cup (8 tablespoons) of butter with brown sugar and corn syrup over medium heat. Stir until smooth and pour into a 13 x 9 x 2 - inch inch baking pan.
Cut the Hawaiian bread into 1 - inch slices. Arrange the bread slices in one layer in the baking pan. Whisk together eggs, whipping cream, rum extract, and salt in a bowl until well combined and pour over bread. Cover and chill for at least 8 hours.
Remove casserole from refrigerator and stand at room temperature for 10 minutes. Preheat oven to 350 degrees F. Bake, uncovered, about 35 minutes or until bread is puffed and edges are golden. Let stand for 10 minutes.
While casserole is baking, prepare sauce. In a blender or food processor puree strawberries to desired consistency. In a small saucepan stir sugar, water and cornstarch. Cook over medium to high heat until thickened and bubbly. Add pureed berries and cook for 10 minutes more. Stir in 1 tablespoon butter.
Serve French toast casserole warm topped with strawberry sauce.