I'm a WordPress Multi-Site trouble-shooter. I have Better WP Security running successfully on Multi-Site.
I've implemented most of the things in the plugin on my own (the plugin is Much Easier!) I haven't seen anything this plugin does that would block Multi-Site in general; but could be something specific to your hosting configuration.
Since you are running into problems simply activating the plugin, which doesn't enable most of the security improvements until you deliberately turn them on, the problem might not be with the plugin but with server permissions or with something else that will take troubleshooting your setup, not a problem with the plugin.
"Spam user registration" is turned off in My Sites, Network Admin (wp-admin/network/) and then Settings, Network Settings. I set "Allow new registrations" to "Logged in users may register new sites" but you might for now want "Registration is disabled".
Manually delete all unwanted Users (My Sites, Network Admin, Users)
Wild guess -- are you able to install other plugins? BackWPup (you have backed up your database, right???) or Akismet (blocks spam comments). There could be a problem from shifting your Multi-Site to a different server, and you're just seeing it as you try activating this plugin...