I wish I could help more, but the old “side jobs are forbidden” being in the company handbook is legally flimsy, if not completely unenforceable. Just because your company says so, or it’s in your contact doesn’t actually make it legal.

The MHLW states that it actually has to provably interfere with your job in some way. The hours outside your day job are your own, not your companies. Don’t close your kojinjigyo and just submit your own tax adjustment every year. Unless you’re making lots of money your HR won’t even know.