This is an interesting challenge.
If the site was dynamic with a backend, and you had SSO between Discourse and site this would have a clear path forward. Your backend know who the user is, and can use an Admin API Key to query Discourse notifications for the current user and send it forward with the page on every request.
Since site is a static page, it doesn’t really have a logged-in state right?
We have the option to Embedding a list of Discourse Topics in another site and Embedding Discourse Comments via Javascript both which works with static sites.
I think having a new one for notifications is a great idea!