Make tabbouleh: Pulse cauliflower in a food processor until crumbly, about 10 times. Transfer to a bowl. Add parsley, mint, 2 tablespoons of the oil, 2 tablespoons of the lemon juice, 1/2 teaspoon of the pepper, and 1/4 teaspoon of the salt to bowl; stir until combined.
In a separate bowl, combine tomatoes, cucumber, chickpeas, onion, olives, and remaining 1 1/2 tablespoons oil, 2 tablespoons lemon juice, and 1/4 teaspoon each salt and pepper; toss to coat.
Place 3/4 cup tomato mixture in center of each of 4 plates. Divide tabbouleh, baba ghanoush, artichokes, and bell peppers evenly among plates, spooning around tomato mixture. Drizzle with additional oil, if desired. Serve immediately with pita rounds, if desired, and lemon wedges.