In a cup, sprinkle the gelatin over 1/4 cup of water; let stand until softened, 5 minutes. Combine 1/2 cup of the sugar and the ginger in a food processor and process until the ginger is finely chopped, then transfer to a small saucepan. Add the milk and salt and cook over medium heat, stirring occasionally, until simmering, about 5 minutes. Remove from the heat and stir in the softened gelatin. Stand the pot in a bowl of cold water to cool to room temperature, about 30 minutes.
Place the sour cream in a large bowl; gradually whisk in the cool, but not set, ginger and milk mixture until smooth. Cover with plastic wrap and refrigerate the mixture until cold and thick, up to 1 hour.
Meanwhile, combine the remaining 1/2 cup of sugar and 3/4 cup of water into a small saucepan and bring to a boil over medium heat, stirring, until the sugar dissolves. Remove from the heat and let cool. Stir in the rum.
Beat the heavy cream with an electric mixer until stiff and then gently stir into the cold, but not set, ginger mixture. (If the ginger mixture has begun to set, whisk it briefly to soften it before adding the cream.)
Reserve 1 cup of the raspberries and 1 cup of the ginger cream and set aside. Place one-third of the cake cubes in an even layer in a 3-quart footed trifle bowl. Spoon a third of the rum syrup over the cake cubes. Slice 1 banana over the cake and scatter half of the remaining raspberries over the bananas. Spoon half of the remaining ginger cream over the fruit. Repeat the layering of cake, syrup, fruit and cream. (You should have enough cake and syrup for one more layer.) Place the remaining cake on top and drizzle with the last of the syrup. Spoon the reserved cream into the center and garnish with the reserved berries. Chill the trifle until ready to serve (3 hours and up to overnight).