This is an extreme remedy but...
If your host won't let you pull the switch, you can cut back on your own. Every night before you go to bed, upload an htaccess file that contains some kind of ungrammatical garbage. While you sleep, everyone will get a 500 error. This is not a pretty solution. But you can make a custom 500 page for your human visitors, something like "Closed for maintenance-- check back in a few hours."
Or put in a "Deny from all" directive and comment it out every morning. There are lots of other ways to lock people out if you are willing to be brutal and unkind about it :(
This is assuming you are allowed to make your own htaccess and look at your own raw logs. If not, you may as well change hosts anyway ;)