Preheat a camp oven to 350 degrees F (177 degrees C). Alternatively, you can use a closed grill, smoker, or charcoal grill set-up for indirect cooking.
Mix together the milk, white sugar, brown sugar, vanilla, rum, eggs, and cinnamon. Whisk together until the sugar has dissolved then add the bananas.
Pour the mixture over the cubed bread and mix with your hands until the bread is completely wet.
Fill a greased skillet with the mixture, leaving no air gaps. Generously sprinkle the turbinado sugar over the top of the mixture in the pan, if using.
Place the skillet with the bread pudding into the preheated camp oven. Cook for about 1 hour. Test for doneness with a toothpick, or when it reaches a 160 degree F (71 degrees C) internal temperature. Remove from the camp oven and let cool.
Banana Sauce
In a skillet, add the butter, brown sugar, dark rum, and vanilla. Simmer until the butter is melted and the brown sugar is dissolved. Keep stirring to prevent the sauce from separating. Add the banana slices and cook until they are tender and covered with the sauce.
To Serve
Spoon the warm banana pudding into a bowl then top with the banana sauce.