Position a rack in the middle of the oven and preheat to 350 degrees F. Line a 9-inch-square baking pan with aluminum foil, leaving a 2-inch overhang on all sides. Lightly spray the foil with cooking spray.
Pulse 1 cup pretzel pieces in a food processor until finely ground (you should have about 1/2 cup crumbs). Transfer to a bowl. Whisk in the flour, baking powder and salt.
Melt the butter in a saucepan over medium heat. Remove from the heat and stir in the brown sugar. Let cool slightly, then stir in the eggs and vanilla. Stir the pretzel crumb mixture into the butter mixture in two additions.
Spread the batter in the prepared pan. Scatter the remaining 1/2 cup pretzel pieces on top. Bake until golden and a toothpick inserted into the center comes out clean with a few crumbs, 25 to 30 minutes. Let cool completely in the pan on a rack, then lift the foil to remove the bars from the pan and transfer to a baking sheet.
Preheat the broiler. Press the marshmallow halves, cut-side down, in even rows on top of the bars. Broil, rotating the baking sheet as needed, until the marshmallows are golden brown, 1 to 2 minutes; let cool.
Put the peanut butter chips and 1/2 teaspoon vegetable oil in a microwave-safe bowl and microwave in 30-second intervals, stirring, until melted and smooth. Repeat with the chocolate chips and the remaining 1/2 teaspoon vegetable oil in another bowl. Drizzle the melted peanut butter and chocolate mixtures over the bars. Let harden at room temperature or chill to set before cutting.