I am not sure how the cost-sharing deal would work out.
One complicating factor is that U.S. put leash on S. Korea's ballistic missile and nuclear program. If S. Korea is asked to pay the full expense, they probably want those restriction to be lifted. Especially ballistic missile program. Currently their ballistic missile range is capped at 800km, IIRC.
North Korea ‘prepares submarine attack’ on South Korea as crisis talks enter third day: live
Seoul refuses to back down and demands apology from Kim Jong-un for landmine blast it claims maimed two soldiers - follow the latest updates