Open beer can; drink half. Carefully pierce top of beer can with "church-key" can opener several times; set aside. To prepare chicken, soak wood chips in water 1 hour. Combine salt, sugar, paprika, and pepper; set aside.
To prepare grill for indirect grilling, place a disposable aluminum foil pan in center of grill. Arrange charcoal around foil pan; heat to medium heat.
Remove and discard giblets and neck from chicken. Rinse chicken with cold water; pat dry. Trim excess fat. Starting at neck cavity, loosen skin from breast and drumsticks by inserting fingers, gently pushing between skin and meat.
Rub 2 teaspoons spice mixture under loosened skin. Rub 2 teaspoons spice mixture in body cavity. Rub 2 teaspoons spice mixture over skin. Slowly add remaining spice mixture to beer can (salt will make beer foam). Holding chicken upright with the body cavity facing down, insert beer can into cavity.
Drain wood chips. Place half of wood chips on hot coals. Coat grill rack with cooking spray. Place chicken on grill rack over drip pan. Spread legs out to form a tripod to support the chicken. Cover and grill 2 hours or until a meat thermometer inserted into meaty portion of thigh registers 180°. Add remaining wood chips after 1 hour and charcoal as needed.
Lift chicken slightly using tongs; place spatula under can. Carefully lift chicken and can; place on a cutting board. Let stand 5 minutes. Gently lift chicken using tongs or insulated rubber gloves; carefully twist can and remove from cavity. Discard skin and can.
To prepare sauce, combine cola and remaining ingredients in a saucepan; bring to a boil. Reduce heat, and simmer 6 minutes. Cool. Serve with chicken.