You're assuming Satoru's mom's phone has a lock. I don't have one. The killer wouldn't be able to text using the phone if it was locked.

But even if it was, I'm pretty sure it's not hard to find someone who can crack a celphone lock. People do that for a few bucks in my country. Maybe the killer can even do it himself from net videos lol. You can just crack it at a temp location, turn it off again, and move somewhere else. Then turn it on and send the message. Japanese police, or most local police for that matter, aren't the FBI.