Preheat the oven to 475 degrees F with a rack on the lowest rung.
Drizzle the oil in a heavy, rimmed half-sheet pan and turn to coat the bottom. Add the dough and press and stretch to the corners with your fingers. (If the dough doesn't go all the way to the corners, let it rest for 5 minutes, then once it has relaxed a bit, stretch again.)
Combine 1/2 cup each marinara and barbecue sauce in a small bowl. Put the chicken in another small bowl and add the remaining tablespoon each of marinara and barbecue sauce. Toss to coat the chicken in the sauce.
Spread the BBQ marinara mixture on the pizza crust, leaving about 1/2-inch border around the edge. Sprinkle with 1 cup each Cheddar and pepper jack. For the meat side, top with the chicken, ham, bacon and pepperoni. For the vegetable side, top with the bell pepper, zucchini and tomato. Scatter the red onion over both sides. Sprinkle both sides with the remaining 1/4 cup each Cheddar and pepper jack.
Bake the pizza on the bottom rack until the crust is golden on the bottom and the top is browned and crusty, 25 to 30 minutes. Let cool in the pan for 5 minutes before cutting into squares to serve.