How to push data from server application to a manifest v3 browser extension in Edge?

Question

We maintain a few browser extensions which currently rely on long-running connections to our cloud services to receive real-time data. In manifest v3 extensions, long-running connections no longer work because the service worker can be shut down regularly by the browser. As an alternative, is there some way that our cloud service could push data or notifications to the extension, such that the service worker will be activated (if necessary) to handle it?

We've tried looking at the Web Push API, but it doesn't seem to be available from within an extension service worker.