Thanks @baris for the very helpful code suggestion! We ended up including additional CSS directly into our own theme, as it looks like some other themes cannot be easily modified by skins anyway.

Thanks @baris for the code snippet. I had to modify it a bit to have it run only after the DOM is fully loaded as otherwise app.require wasn't available:

<script>
document.addEventListener("DOMContentLoaded", async () => {
  if (typeof app !== 'undefined' && typeof app.require === 'function') {
    const myWidget = await app.require("mywidgetmodule");
	myWidget.init();
  } else {
    console.error("app.require is not available.");
  }
});
</script>

If you have suggestions how to do it better feel free to tell me. However, my problem is solved, so thank you!