Granted, such a secret program wouldn't be much of a "MAD"-style deterrent, but they would give Japan the the option of taking matters into their own hands if NK flips out and a limp-wristed (read Democrat) US administration refuses to help outside of taking loud public umbrage. Nuclear SLBMs, combined with Japan's avowed spy satellite capabilities might allow for a quick knockout strike against NK WMD assets in the event of imminent hostilities.
If it were easy to take out NK nukes, it could be done conventionally. If it is hard and messy, they either wouldn't sign off on it or would expect us to do it. They need a deterrent to China and Russia in any event, if they do such a thing. If they don't trust the US umbrella for one, why would they trust it for the other? They'd need their own, if they wrote us off as unwilling. When they do so you will know it, because they will hold a press conference and tell China what they can do if they have to.