Here is the statute dealing with having a firearm while archery hunting:
70-4-123. Hunting with bow and arrow while in possession of firearms or accompanied by a person in possession of firearms -- Penalty.
(a) It is unlawful for any person hunting big game with a bow and arrow to be in possession of any firearms or be accompanied in hunting by any person possessing firearms during the archery-only deer season; provided, that persons authorized to carry a handgun pursuant to § 39-17-1351 may carry a handgun as defined in § 39-11-106(a) while hunting big game with a bow and arrow during the archery-only deer season.
(b) A person who violates this section commits a Class C misdemeanor.
HISTORY: Acts 1967, ch. 40, §§ 1, 2; 1982, ch. 738, § 23; T.C.A., § 51-443; Acts 1989, ch. 591, § 113; 2010, ch. 621, § 1.
Without a carry permit, it would be illegal to have a centerfire weapon during the muzzleloader season.
There are some gray areas concerning possession of weapons not legal for hunting on a WMA, so your best bet is to discuss the WMA situation with Region 4.