I don't think SK will choose this path, but If SK choose to do so, it won't be limited to a BMD program. It will become a mini arm race directly against NK, but also alarms Japan to join in because Japan will not trust a militarily independent SK from US. Japan's move will then trigger China and Russia to step up. It is a chain reaction that SK will loose even more than not to do so.
But on the evil side of the thought, pulling more money from maintaining SK's tech advancement into spiraling military cost does serve China well. When there is no Samsung (or the likes) around, SK will be more cooperative or less of a concern to China.
Note that both SK and Israel are spending a similar amount of technology R&D (4.3% of GDP)
So SK doubling military spending to Israeli levels should be doable.
And an independent SK should lean towards China, and which would cause Japan to panic and militarise anyway. Russia wouldn't care at all