Make the cake: Preheat the oven to 350 degrees F. Butter two 8-inch round cake pans and line with parchment paper. Butter the parchment and dust with flour.
Spread the pecans on a baking sheet and bake until fragrant and toasted, about 8 minutes. Let cool slightly, then roughly chop. Toss with the bananas, pineapple and 1/2 cup flour in a small bowl.
Whisk the remaining 2 1/4 cups flour, the cinnamon, nutmeg, ginger, baking soda and salt in a bowl. Beat the eggs and granulated sugar in a separate bowl with a mixer on high speed until thick and light, 5 minutes. Gradually beat in the vegetable oil.
Sprinkle the flour mixture over the egg mixture, then gently fold to make a thick batter. Fold in the pecan-fruit mixture, then transfer the batter to the prepared pans. Bake until the cakes are firm and a toothpick inserted into the middle comes out clean, 50 to 55 minutes. Cool in the pans on a rack, 25 minutes, then invert the cakes onto the rack to cool completely.
Make the frosting: Beat the cream cheese in a large bowl with a mixer until fluffy, then gradually beat in the butter until combined. Sift the confectioners' sugar over the cream cheese mixture and beat until smooth. Add the lemon zest and vanilla and beat until light and fluffy.
Place one cake layer on a serving plate. Spread about half of the frosting on top, then cover with the other cake layer. Spread the remaining frosting over the top and sides of the cake.