Preheat oven to 325°F. Melt 2 tablespoons of the unsalted butter in a small saucepan. Using a silicon pastry brush, 'paint' the Bundt pan with half the melted butter. Use a small fine mesh sieve to dust the flour inside the Bundt, tapping it so it is well floured. Tap out any excess. Set aside.
To the remaining half of melted butter, add the chopped pecans and cook over medium heat for 1 minute. Scrape the buttered pecans into the bottom of the Bundt pan. Wipe out saucepan and set aside for use for the glaze.
In the bowl of a stand mixer with the paddle attachment (or a large mixing bowl using handheld electric mixer), combine the dry cake mix, dry coconut cream pudding mix, 1/2 cup coconut rum, 1/2 cup tap water, the vegetable oil, and the 4 eggs. Mix for 2 minutes on medium speed, then mix in 1/2 cup of the flaked coconut.
Pour batter into prepared Bundt pan and bake for 1 hour. Test for doneness by inserting a toothpick into center. If pick comes out clean, the cake is ready.
While the cake bakes, prepare the glaze. Melt the remaining unsalted butter in the saucepan, then add sugar and water. Bring mixture to a low boil for 5 minutes, whisking constantly. NOTE this sugar mixture will be VERY HOT and is prone to BOILING OVER QUICKLY, so do not walk away from the stove even if only for a moment. Continue with the glaze by removing from the heat and add rum, whisking in well.
Remove the cake from oven and immediately poke lots of holes all over the top of the cake plunging down about 3/4 of the depth of the cake using a bamboo kebab. Ladle half of the hot glaze onto cake over the tiny holes and around the outside edges where the cake has pulled away from the pan. Cool cake for 1 hour in the Bundt pan. NOTE I like to continue to ladle additional glaze over the tiny holes during the hour-long cool, as I much prefer to have the crumb saturated on the inside versus glazed on the outside. It's okay to use some of the remaining glaze to do this, or to simply keep the remaining half of glaze to finish the cake.
Invert cake onto cake pedestal or cake plate. GENTLY tap the pan to loosen the cake. Use a butter knife to gently loosen any pecans that remain in the Bundt and sprinkle over top the cake. Pour other half (or whatever amount remains) of glaze over cake once it has been inverted onto a cake plate, allowing it to slowly drip down the sides.
Garnish with remaining flaked coconut and serve.