Is this a good reason to put a dog down?

Sorry, but uming you've described the situation accurately, your mother's not fit to have the care of any dog at all. You'd all be best off to find good homes for all three dogs and don't get any more as long as she's in charge of them. Or at least take Kioko to a shelter or a Shiba Inu rescue group instead of putting her down. And buy lots of insurance, because when untrained, unsocialized Lola goes too far one day soon, you're going to need it.