Stir together preserves, ginger and mustard in a small bowl. Set aside 1/4 cup; put remaining mixture in a medium bowl.
Add peaches, bell peppers and scallions to medium bowl with preserves mixture. Stir gently. Stir in basil and lime juice. Cover and chill until ready to serve.
Put chicken on a foil-lined baking sheet in a single layer. Squeeze lime halves over both sides of chicken; season with salt and pepper.
Turn on all grill burners and preheat to medium-high (350°F to 400°F). When grill is ready, turn one burner off for indirect heat and leave others on for direct heat. Brush grill with oil. Place chicken on grill rack over direct heat. Cover and grill for 5 minutes on each side. Baste chicken with remaining preserves mixture and move to grill rack over indirect heat. Cover and grill for 10 minutes longer, turning once and basting often. Remove chicken from grill; let stand 5 minutes before serving. Serve with reserved peach relish.