I have no experience in this matter but my first thought is that when you detect that the internet connection is down, redirect DNS requests to a DNS server that maps all requests to a web server that says "the internet is temporarily down" (e.g. set up your own DNS server and have your DHCP server deliver that, possibly forcing renewal of all leases during change over). Of course this is only visible when web browsing but that will catch the vast majority of cases; for non-web connections the customer will likely notice that it isn't working and hopefully try to access the web. The other down side is it won't have an effect on customers that override their DNS settings (e.g. I always use Google's public DNS), but I think this is a fair minority as well. I guess there might be some issues with client-side DNS caching as well.
You probably don't want to change the SSID for a few reasons, including but not limited to:
The internet being down/up will not be transparent to users.
Users will lose their wifi connection if it switches over.
Users won't automatically connect to "~local only~" unless they choose to, and I know I wouldn't choose to because it's a network name I don't recognize (and doesn't look like something I'd want to connect to at e.g. a coffee shop anyways).
Users likely won't even make the association between your public network and "~local only~" (or necessarily know what "local only" even implies). From their point of view, the public network is gone and, independently, a new unrecognized network is present.
It's kind of sloppy (IMHO) and might not be a great PR move for your system.
You could also attempt to let users know in other ways that the connection is generally unreliable, e.g. if you have a web site that describes your service.
Additionally, it seems to me that these days wifi is ubiquitous enough that if the web browser is timing out making requests, they'll generally come to the natural conclusion that "the internet connection isn't working right now" without you having to tell them - so you could do nothing -- but I do think redirecting them to a page that says the service is down will give them the best experience possible.
I think this is a different technique than the captive portal discussed in the comments, as it's out of the picture entirely when the internet is up and does not involve manipulating data in place. However, if you also can't set up your own DNS server for some reason, I would definitely just do nothing rather than switching the SSID, which is essentially the same thing anyways if you consider what the general response of a user would be when faced with your public SSID no longer being present and an unrecognized new one named "~local only~".