japan conducting a "surgical strike" would be the last thing i would expect. when was the last time we saw japan take any military action? Japan is militarilu powerful nation, but self defence force is not USAF and south korean airforce is not serbian airforce.
suppose Japan gives military support to North Korea and destroy south korean military assets. Now we have a unified korea under north korean government. this is the scenario you are suggesting when japan attacks south korea. furthermore, a south korea facing a japanese attack will have no choice, but to ally with China. Now you will have Chinese warships cruising up and down japanese coasts.
the best thing japan can do is what japan has been doing for the past sixty years. which is do nothing and let the violence not spill over into japan.