Preheat oven to 325 F.
In a large mixing bowl, place flour, sugar, pudding mix, baking powder, salt, butter, and oil and mix at mediu speed until everything is thoroughly incorporated and the mixture looks like sand.
Beat in the milk. Add the eggs one at a time, beating well between each addition. Scrape the bowl thoroughly and beat briefly to incorporate.
Stir in the rum, vanilla, and butter-rum flavoring.
Prepare Bundt pan with cooking spray. Pour batter into prepared pan and spread it level with a spatula
Bake for 50- to 60-minutes. Test for doneness using a skewer, toothpick, or dry piece of ucooked spaghetti. It should come out clean when inserted into the center area of the cake. Do not overbake. Remove cake from the oven. Leave the cake in the pan while you prepare the syrup.
In a medium-size saucepan, combine the butter, water, sugar, salt, and rum. Bring to a rapid boil then reduce to a simmer to cook—without stirring—for 5 to 8 minutes, until the syrup thickens slightly. Remove from the heat and stir in the vanilla.
Using a long skewer or similar, poke holes all over the cake. Pour about 1/4 cup syrup over the cake, while it is still in the pan. Allow the syrup to soak in, then repeat again and again, until all the syrup has been absorbed. See NOTE 4.
Cover the pan loosely with plastic wrap, allowing the cake to sit overnight at room temperature to cool completely and for the syrup to soak into all of the cake.
When ready to serve, invert the cake over a serving plate. If it won't loosen to fall out of the pan easily, do not force it. Instead, tent the pan with foil, place it in the oven on the bottom rack and turn it on to 350 F. for 10 minutes. Remove the cake from the oven, and now tip it onto the serving plate. Serve and enjoy!
If any of this amazing cake remains, put it in a cake saver OR wrap it securely in plastic wrap, storing it at room temperature for 2 to 3 days. You may confidently freeze for longer storage, up to one month.