If you need to prevent showing the new message to the users who hid the old one, how about prevent updating the id when you edit MediaWiki:Sitenotice as "minor"?

That sounds like a not-bad idea, but still, I bet admins would be confused a bit. It would need some docs inlined into the edit form, and, if we were to do that, we may as well just put there a reminder to increment Sitenotice id. And such a reminder is just what we have on Nonsensopedia: MediaWiki:Editnotice-8-Sitenotice lets you set up a notice appearing above the edit form. In our case it's something like this (Polish): https://nonsa.pl/wiki/MediaWiki:Editnotice-8-Sitenotice

Idk, that works good enough for us, but I admit isn't perfect. I have mixed feelings about the minor edit feature :)