If you got into an accident after the 30days, I'm sure they would be happy to tell you to screw yourself. Insurance companies will use anything to get out of paying.
Jessica's finger is right on the The Button. Insurance:
How would you convince the "…state arizona Quebec b.c. etc" insurance company that you're the sort of risk they can make money from when you don't even reside in their territory? They can't seize the car or any of your assets, can't even assess or appraise them, and they'd be paying out mechanics and shops they don't know and can't evaluate either.
They might just see your scheme as a way to scam them for insurance falsely obtained. Even if you went local, it's unlikely they'd be impressed with how clever the phony registration was. More likely they'd just say, "Sorry, we don't need your biz".
IF you found BC and Quebec rates that attractive, support politicians with public auto insurance in their party platforms, like their voters did.