Preheat grill to medium heat (300ºF to 350ºF). In a large bowl, toss zucchini, squash, bell pepper, onion and Italian seasoning with 1 Tbsp. olive oil. Lightly brush grill rack with oil. Grill vegetables, covered, until lightly charred in spots but still crisp-tender, 4 to 6 minutes on each side. Remove vegetables, let cool and chop. Cover and set aside.
Bring a pot of salted water to a boil. Cook pasta until al dente, about 10 minutes or as package label directs. Drain.
Toss pasta, vegetables, beans, pesto, cheese, 1/2 tsp. salt and 1/2 tsp. pepper to combine. Cover and refrigerate until thoroughly chilled, at least 2 hours. Drizzle with remaining olive oil and lemon juice just before serving.