Make the shortcakes: Cook the bacon in a large skillet over medium heat, turning occasionally, until crisp, 9 to 11 minutes; drain on paper towels. Finely chop 2 slices of the bacon and roughly chop 4 slices; set aside.
Preheat the oven to 375 degrees F and line a baking sheet with parchment paper. Pulse the flour, baking powder, granulated sugar and salt in a food processor to combine. Add the butter and pulse until the mixture looks like coarse meal. Add the milk, vanilla and finely chopped bacon and pulse until the dough just comes together. Turn out onto a lightly floured surface and pat into a 5-by-7 1/2-inch rectangle.
Cut out 6 rounds with a 2 1/2-inch biscuit cutter; arrange about 2 inches apart on the prepared baking sheet. Brush the tops lightly with milk and sprinkle with granulated sugar and the chopped peanuts. Bake until the shortcakes are puffed and golden brown, 15 to 18 minutes. Transfer to a rack to cool completely.
Meanwhile, make the whipped cream: Beat the heavy cream, confectioners' sugar, peanut butter and vanilla in a large bowl with a mixer on medium-high speed until soft peaks form; refrigerate until ready to use.
Make the caramel: Combine the granulated sugar, 3 tablespoons water and the cream of tartar in a medium saucepan. Cook over medium-high heat, swirling the pan occasionally but not stirring, until the mixture is light amber, about 7 minutes. Remove from the heat and carefully stir in the heavy cream and rum (the mixture will bubble). Return to medium heat and simmer until slightly thickened, about 1 minute; add the bananas and roughly chopped bacon.
Split the shortcakes. Top with the banana mixture and peanut butter whipped cream.