Cut 1/2-inch deep slashes in the skin-side of each chicken half: two cuts in each breast, two in each thigh, and one in each leg. Remove wing tips.

Whisk vinegar, 2 tablespoons barbecue sauce, and garlic together in a large bowl. Place chicken in the bowl and turn to coat. Arrange chicken halves in the bottom of the bowl with the cut sides down; cover with plastic wrap and refrigerate for 1 hour.

Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Remove chicken from the bowl and pat dry with paper towels; discard any remaining marinade. Place chicken halves, skin-side up, on a plate and season with salt, pepper, paprika, onion powder, and cayenne pepper.

Cook chicken, skin-side down, on the preheated grill until grill marks appear, 3 to 4 minutes. Turn chicken over, close the grill lid, and cook, basting with remaining barbecue sauce every 6 minutes, until no longer pink at the bone and the juices run clear, about 35 minutes. An instant-read thermometer inserted into the thickest part of the thigh, near the bone should read 165 degrees F (74 degrees C).