Using importScripts() in a service worker in Firefox under Manifest V3

I’m working on a browser extension. In Chrome, I can simply do this:

manifest.json:

"background": {
        "service_worker": "scripts/vh_service_worker.js",
},

and in my service worker, I can use ìmportScript("somefile.js") without issues.

In Firefox, background.service_worker is not implemented for Manifest v3.

Attempt #1:
I load the service worker as background script:

manifest.json:

"background": {
        "scripts": ["scripts/vh_service_worker.js"]
    },

This generally works well, but unfortunately it does not seem to be executed in the WorkerGlobalScope, so the function importScripts() is not available.

Attempt #2:
I tried to use the background script to register a service worker manually:

manifest.json:

"background": {
        "scripts": ["scripts/vh_service_worker_loader.js"]
    },

vh_service_worker_loader.js:

if ("serviceWorker" in navigator) {
    navigator.serviceWorker
        .register("scripts/vh_service_worker.js")
        .then((registration) => {
            console.log("Service Worker registered with scope:", registration.scope);
        })
        .catch((error) => {
            console.error("Service Worker registration failed:", error);
        });
}

But unfortunately there is no navigator.serviceWorker entry available, so the if condition always return false.

So my question is:
How can I use importScripts() in Firefox for an extension using Manifest V3 ?