Sounds cool! Consider contributing it back to GitLab as a project service. Make sure to mention this comment in your merge request.

Actually it's just one <script> header, and a single (ugly) <button> tag duct-taped right into the page template.

I don't think it is either upstream-ready or especially advantageous for the project. Am I wrong?

If you like it someone else might. And submitting it spares the you trouble of maintaining a fork when you upgrade. But making it a project service might be hard. So feel free to not submit it. Alternatively you could write a small blog post about it, maybe it inspires someone else to contribute.