Preheat the oven to 350 degrees F (175 degrees C). Spray a 9x12-inch glass baking dish with cooking spray.

Spread bread cubes on an ungreased baking sheet.

Bake in the preheated oven, stirring once or twice, until toasted, 5 to 7 minutes. Remove from the oven and set aside.

Whisk eggs, sugar, and 1/8 teaspoon salt together in a large bowl until blended.

Bring cream and half-and-half to a simmer in a medium saucepan. Remove from heat. Pour some of the hot cream mixture into the egg mixture, whisking quickly to avoid scrambling the eggs. Continue adding cream mixture slowly and whisking until all has been added. Whisk in vanilla extract and set custard aside to cool.

Combine brown sugar and butter in a large skillet over medium-high heat; whisk until butter is melted and mixture is smooth. Whisk in rum, lemon juice, and 1/4 teaspoon salt. Add bananas and toss just until coated. Remove from heat.

Toss 1/3 of the toasted bread in the prepared baking dish and top with 1/2 of the banana-syrup mixture. Layer with 1/2 of the remaining bread, the remaining banana-syrup mixture, and the remaining bread. Pour custard over the top and allow pudding to stand for 30 minutes, pressing down on the bread occasionally so it absorbs the custard.

Cover bread pudding with aluminum foil and place the baking dish inside a large baking pan. Fill the large pan with warm water so it reaches halfway up the sides of the pudding dish.

Bake in the preheated oven for 30 minutes. Uncover and bake until pudding has puffed up and the center is set, about 45 minutes longer. Remove from the water bath and let stand for 15 minutes.

Meanwhile, pour powdered sugar evenly into a heavy saucepan. Cook, without stirring, over medium-high heat until sugar begins to melt and caramelize on the bottom, 3 to 4 minutes. Stir until all sugar is melted and smooth. Add 1 cup cream, rum, and salt; stir until smooth. Stir in remaining 1/4 cup cream. Remove from heat and stir in butter until caramel sauce is smooth. Serve with bread pudding.