Place oven rack on lowest shelf. If you have a stone, place it on rack. Preheat oven to 550°F, or your oven's highest temperature, for 45 minutes.
Warm 1 Tbsp. oil in a medium nonstick skillet over medium-high heat. Add mushrooms and stir to coat. Season with salt and pepper. Cook, stirring occasionally, until mushrooms are golden, about 4 minutes. Add garlic and bell pepper and sauté until pepper slices are crisp-tender, about 30 seconds. Transfer to a plate to cool.
While vegetables are cooking, dust a 14-inch-wide pizza peel with cornmeal, then flatten dough on top. Using lightly floured fingers or a floured rolling pin, stretch dough into a 13- to 14-inch round. (Sprinkle peel with more cornmeal if necessary to make sure crust slides around easily on peel; otherwise it will stick when you try to transfer it to oven.) Brush remaining 1 Tbsp. oil over surface of crust.
Spoon sauce in a spiral over pizza, leaving a 1-inch border. Scatter vegetable mixture, olives and capers over pizza. Top with cheese.
Carefully slide pizza onto stone. Bake until crust is just slightly charred on top and golden on bottom (lift an edge to check) and cheese is melted and bubbling, about 8 minutes. Using peel, transfer pizza to a cutting board. Let pizza rest for a minute or 2, then slice and serve.