Preheat grill for medium heat and lightly oil the grate.
Combine shrimp, 1 tablespoon soy sauce, and lime zest in a bowl. Allow shrimp to marinate at room temperature while preparing remaining ingredients.
Bring a large pot of water to a boil. Add rice stick noodles to boiling water and remove from heat. Cook until noodles have cooked through, but are still firm to the bite, 3 to 5 minutes. Rinse will cold water until chilled; drain well.
Whisk lime juice, remaining 3 tablespoons soy sauce, honey, sesame oil, remaining 1 teaspoon lime zest, anchovy paste, and hot pepper sauce in a large bowl. Add peaches, cabbage, carrots, cilantro, basil, and garlic and toss to coat. Stir rice noodles into vegetable mixture until combined; set aside.
Thread about 8 shrimp on 4 wooden skewer by piercing shrimp through the tail and the upper body. Pierce cherry tomatoes onto remaining 2 skewers.
Place shrimp and tomato skewers on preheated grill. Cook tomatoes until they begin to split, about 3 minutes per side; transfer to a platter. Continue to cook shrimp until they are bright pink on the outside and is no longer transparent in the center, 2 to 3 minutes more. Remove tomatoes and shrimp from skewers.
Transfer noodle salad to a large platter. Surround salad with tomatoes and shrimp; drizzle any remaining lime dressing over shrimp.