Preheat the oven to 350°F (175°C), with a rack in the middle. Thoroughly coat a 12-cup Bundt pan with baking spray containing flour, including the center tube and all grooves. Alternatively, grease with butter, dust with flour, and tap out the excess.
Whisk the flour, baking powder, and salt together in a bowl. In a separate measuring jug, combine the milk and the 1/2 cup of rum reserved for the cake.
Beat the softened butter and sugar on medium speed for 3–5 minutes, until noticeably lighter and fluffy. Scrape down the sides and bottom of the bowl.
Beat in the eggs one at a time, mixing until each is incorporated before adding the next. Mix in the vanilla.
On low speed, add the flour mixture in three portions, alternating with the milk-rum mixture in two portions. Begin and end with the flour. Mix only until incorporated, then fold briefly with a spatula to check for unmixed flour at the bottom.
Spoon the batter into the prepared pan and smooth the surface. Bake for 50–60 minutes, checking at 45 minutes. The cake is ready when a skewer inserted halfway between the center tube and outer edge comes out clean or with a few moist crumbs, but no wet batter.
During the final 10 minutes of baking, prepare the syrup. Combine the butter, sugar, and water in a small saucepan. Bring to a gentle boil, stirring to dissolve the sugar, then simmer gently for 2–3 minutes.
Remove the saucepan from the heat and move it away from any flame. Let the bubbling settle, then slowly stir in the remaining 1/4 cup rum and the pinch of salt. Do not ignite the syrup.
Place the baked cake, still in its pan, on a wire rack for 10 minutes. Gently loosen any accessible cake edges if needed, then invert onto a rimmed serving platter and lift away the pan.
While the cake is warm, make small holes over the top with a thin skewer, reaching into the crumb without piercing completely through the cake.
Slowly brush or spoon the warm syrup over the top and sides in several passes, allowing each application to absorb before adding more. If syrup collects on the platter, spoon it back over the cake gradually. Pause if the surface becomes saturated.
Let the cake finish cooling and absorbing the syrup before slicing, allowing about 2 hours total from the time it leaves the oven. Serve in modest slices.