Preheat the oven to 350 degrees F. Line a 9x13-inch baking dish with parchment paper. Mix the flour, baking powder, and salt in a medium bowl.
In the bowl of your electric mixer, add the softened butter and sugar, beat until light and fluffy, 3 minutes. Beat in the egg and rum, then slowly add the flour mixture and coconut milk until well combined. Scrape the bowl and beat again for 30-45 seconds.
Pour the batter into the prepared baking dish. Smooth it out with a spatula and bake for 22-25 minutes.
Once the pina colada cake is out of the oven and has had time to cool, dump the pineapple in a colander and press with paper towels to remove as much excess moisture as possible. Dry pineapple = better icing.
Place the cream cheese in a clean mixing bowl and beat to soften and remove clumps. Then add the crushed pineapple and beat to combine. Turn the mixer on low and slowly add the powdered sugar, until the consistency is like really thick school glue.
Pour the icing over the top of the bars and sprinkle with toasted coconut. Place in the refrigerator for at least one hour. Once chilled, lift the whole sheet of pina colada cake squares out of the baking dish and cut into squares. The icing will still be be slightly loose and will ooze over the edges. Serve immediately or allow the icing to sit and set all the way.