Cut tomatoes into ½" thick wedges. Cut wedges in half crosswise. Toss tomatoes and ½ teaspoon salt together in a colander set in large bowl.
Drain for 30 minutes. Place sliced thin red onion in small bowl, cover with ice water, and let sit for 15 minutes. Whisk red wine vinegar, oregano, pepper, and remaining ¾ tsp salt together in second small bowl.
Discard tomato juice and transfer tomatoes to now-empty bowl. Drain onion and (set aside, as with bell peppers and olives add to bowl with tomatoes). Add vinegar mixture, cucumber, and capers and toss to gently coat.
Season with salt and pepper to taste. Transfer to serving platter and top with feta. Season each slice of feta with extra oregano to taste. Drizzle feta with remaining 1 Tbsp olive oil.