Place steaks in a resealable plastic bag or glass baking dish. In a small bowl, whisk together steak sauce, tomato sauce, chopped garlic, vinegar, liquid smoke, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Add to steaks in bag or dish; place in refrigerator to marinate for at least 30 minutes.
Heat gas grill to medium or prepare charcoal grill with medium-hot coals. Cut onion into 1/3-inch-thick slices. Secure onion slices with toothpicks to keep rings from separating. Brush with oil and season with remaining 1/8 teaspoon salt and pinch of black pepper.
Once grill is heated, add steaks and cover. Grill 5 minutes. Transfer marinade to small saucepan and bring to a boil. Boil 3 minutes; set aside.
Flip over steaks, and grill an additional 5 to 6 minutes, adding onions to the grill. Grill onions, turning, 4 minutes per side until done.
Remove steaks from grill and let rest at least 10 minutes. Meanwhile, remove toothpicks from onion slices and separate into rings. Serve steaks with onion and mesquite sauce on the side. Makes 6 servings.