Heat oven to 350°F. Line a 9×13-inch pan with parchment, leaving overhang, and lightly grease.
Whisk flour, oats, granulated sugar, brown sugar, baking powder, salt, and cinnamon in a large bowl.
Cut in the cold butter with a pastry cutter until sandy, pea-sized crumbs form. Stir in vanilla.
Press about two-thirds of the crumb mixture firmly into the prepared pan to form an even base.
Combine preserves, raspberries, lemon juice, lemon zest, and cornstarch in a bowl. Gently mash a few berries.
Spread the raspberry mixture evenly over the crust.
Crumble the remaining dough over the fruit, pinching some pieces into larger clumps.
Bake until the top is golden and the filling bubbles at the edges, 33–38 minutes. Cool completely in pan.
Lift out using parchment, dust with powdered sugar, and slice into bars.