Bake the cake:
Position a rack in the center of the oven and heat the oven to 350°F.
Butter the bottom and sides of a 9x13-inch Pyrex baking dish or a nonreactive metal pan. Line the bottom of the baking dish or pan with parchment and lightly butter the parchment.
Sift the flour, baking powder, and salt into a medium bowl and set aside.
Separate the eggs, putting the whites in a medium bowl and the yolks in a large bowl. With an electric mixer, beat the yolks with 3/4 cup of the sugar on medium speed until the mixture is pale and creamy, about 2 minutes. Add the milk and vanilla and beat until combined, 1 minute more.
Clean and dry the beaters and then beat the egg whites, gradually increasing the speed to high, until they reach soft peaks, 2 to 3 minutes. Add the remaining 1/4 cup sugar in a stream, continuing to beat on high, until you reach firm but not dry peaks, 1 to 2 minutes more. Whisk a third of the dry ingredients into the yolk mixture until thoroughly combined. Gently fold in a third of the egg whites with a rubber spatula. Fold in the remaining dry ingredients and egg whites, alternately, in two more batches each, until fully incorporated.
Pour the batter into the prepared dish or pan and bake until a toothpick inserted in the center comes out clean, 20 to 25 minutes. Let the cake cool in the pan on a rack for 10 minutes, then invert the cake onto the rack, remove the parchment, and let cool completely.
Return the cake to the baking dish or pan (the cake will soak up more of the liquid if returned to the pan it was baked in), or invert it onto a rimmed platter.
Soak the cake:
In a 2-quart saucepan, stir together the condensed milk, evaporated milk, coconut milk, and salt until the milks are well blended. Cook over medium-low heat, stirring to avoid scorching, until it begins to bubble around the edges, 3 to 5 minutes. Remove from the heat and pour into a heatproof 4-cup measuring cup.
With a toothpick, prick the cake to the bottom in 1/2-inch intervals. Pour the soaking liquid slowly over the cake, starting at the edges and pausing to let it soak in before adding more. Cover loosely with plastic wrap and refrigerate until the cake is well chilled, at least 2 hours and up to 24 hours.
Top the cake:
In a large bowl, beat the heavy cream with an electric mixer on medium speed. When it begins to thicken, slowly add the rum, sugar and vanilla and continue to beat just until it holds firm peaks, 3 to 4 minutes (be careful not to overbeat). Spread the whipped cream all over the top of the cake, scatter the coconut over the cream and serve.
You can soak the cake in the milk mixture up to a day ahead and top it up to 2 hours ahead.