Place beer, all-purpose flour, whole-wheat flour, salt, oregano, mustard, cayenne and pepper in a blender; blend until smooth, scraping down the sides as necessary. Transfer the batter to a shallow baking dish. Add fish, turning to coat all sides.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Transfer one-third of the battered fish to the pan, placing each piece into a little oil. Cook until golden brown, 3 to 4 minutes per side. Transfer cooked fish to a plate; keep warm. Add 1 tablespoon oil and half the remaining fish to the pan; cook as directed above, reducing the heat if necessary. Cook the remaining fish with the remaining 1 tablespoon oil. Serve immediately.