How do I restrict dashboard access during site maintenance while keeping the public site live?

I have a large multisite install with editors and administrators for a number of subsites. I need to be able to block their ability to make changes to the site while we perform upgrades, etc. I host with WP Engine and have options to create checkpoints or staging sites, but have run into issues with users losing work that they did between the time I started testing upgrades and sometimes reverting back to my checkpoint. I basically need to lock people out of the back-end, but not take down the public website while maintenance is performed.

I've seen the threads for the shutdown plugin, but it appears it's no longer supported. Various maintenance mode plugins I've seen either don't play nice with multisite or disable the front-end of the site during maintenance. Any solutions? I'd imagine anyone managing a large multisite would need something like this.