Thaw fish, if frozen. Rinse fish; pat dry with paper towels. Measure thickness of fish; set aside.
For sauce, in a small skillet cook sweet pepper in 1 tablespoon hot oil over medium heat for 3 to 5 minutes or until tender, stirring occasionally. Stir in chopped tomatoes, 1 tablespoon of the vinegar, the salt, and cayenne pepper. Cook about 5 minutes or until tomatoes are softened, stirring occasionally. Cool slightly. Transfer mixture to a blender or food processor. Cover and blend until smooth. Return sauce to skillet; cover and keep warm.
In a small bowl stir together the remaining 1 tablespoon vinegar, 1 tablespoon oil, and the snipped fresh or dried basil; brush over both sides of fish. Place fish in a greased grill basket, tucking under any thin edges.
For a charcoal grill, grill fish in basket on the rack of an uncovered grill directly over medium coals until fish begins to flake when tested with a fork, turning basket once halfway through grilling. Allow 4 to 6 minutes per 1/2-inch thickness of fish. (For a gas grill, preheat grill. Reduce heat to medium. Place fish in basket on grill rack over heat. Cover and grill as directed.)
Serve fish with sauce. If desired, garnish with cherry tomatoes and fresh basil sprigs.