Within the Constitution of Korea (based on South Korea's IRL), it states that any act of secession is an act that violates the basic and free democratic order. "The Republic of Korea shall seek unification and shall formulate and carry out a policy of peaceful unification based on the basic free and democratic order." - Article 4. The rebellious island would be charged with the following acts of treason: 1) Conspiracy to overthrow the regional government: 2) Insurrection to upend the constitutional law that prevents secession. Possible charges include: 1) Violation of national security. 2) Disruption of national security & public order with the creation of a non-state authorized militia.
---- Preperations ---- To begin preparations for a possible forceful put-down of the rebellious island, 2 fleets would be sent to patrol and blockade the island to prevent possible foreign aid and to restrict their ability to exert force overseas. 3 infantry regiments, totaling ~150K soldiers, would be sent to the most southern point of the Korean peninsula with the explicit orders to shoot and kill any non-civilian landing vessel. 1 armored division would accompany them with very light air support and minimal artillery to prevent any damage to the beaches, cities, or civilian infrastructure. Bombers would be sent over the island to drop tens of thousands of pamphlets demanding that the island either surrender peacefully to remain an integrated region of Korea or face a total military occupation of the island and be seen as a federal district of Seoul.