Bring a large pot of salted water to a boil and cook the cheese tortellini according to package directions until tender. Drain and immediately rinse with cold water to stop the cooking process and cool the pasta completely. Set aside.
While the tortellini cools, chop the salami, pepperoni, and ham into bite-sized pieces. Slice the cherry tomatoes in half, thinly slice the red onion, and chop the banana peppers. If using cucumber, dice it into small pieces for even texture.
In a medium bowl, whisk together the mayonnaise, red wine vinegar, pepperoncini juice, Dijon mustard, olive oil, Italian seasoning, garlic powder, salt, and pepper until smooth and fully combined. Taste and adjust seasoning if needed.
In a large mixing bowl, add the cooled tortellini, chopped meats, mozzarella pearls, tomatoes, onion, banana peppers, and cucumber (if using). Pour the dressing over the top.
Gently toss everything together until all ingredients are evenly coated in the dressing and well combined. Make sure the dressing gets into all the folds of the tortellini.
Cover and refrigerate the salad for at least 30 minutes before serving. This allows the flavors to blend together and the dressing to soak into the pasta.
Right before serving, gently fold in the shredded iceberg lettuce for fresh crunch. Taste and adjust with additional salt, pepper, or a splash of vinegar if needed.