Rinse fish and pat dry with paper towels. Season cavity and outside of each fish with salt and pepper. Place lemon or lime slices and basil leaves in the cavity of each fish. Lay remaining herb branches over each fish and gently tie with kitchen string in three places. Grill immediately or cover and chill the fish up to 2 hours. Transport in an insulated cooler with ice packs.
Thread zucchini pieces and onion wedges on skewers. Cook vegetable kabobs in a small amount of boiling water in a covered 4-1/2-quart Dutch oven for 5 minutes. With tongs, carefully lift out kabobs. Add a mushroom to the end of each skewer. Place in a shallow container; cover and chill up to 2 hours. Transport in an insulated cooler with ice packs.
Combine olive oil and the 1 tablespoon snipped rosemary. Brush fish bundles and vegetables lightly with olive oil mixture. Grill fish and vegetables using direct or indirect heat.
For direct heat:
Grease grill rack. Place fish on grill rack directly over medium coals. Grill about 14 minutes or until fish flakes easily with a fork, turning halfway through grilling. Place vegetable kabobs on grill rack directly over coals for the last 6 minutes of grilling, turning kabobs once.
For indirect heat:
Grease rack of a grill with a cover. Arrange preheated coals around a drip pan. Test for medium heat above the pan. Place fish on the grill rack over the drip pan. Cover and grill about 20 minutes or until fish flakes easily when tested with a fork. Place vegetable kabobs on grill rack directly over coals for the last 6 minutes of grilling, turning skewers once during cooking.
Snip strings and discard herb branches from fish. Serve whole or remove heads and skin. Serve with vegetable kabobs. Makes 4 main-dish servings.