Backport asset path fixes to 15.11 branch

What does this MR do and why?

Backports the fix in this MR to 15.11: !3798 (merged)

We need this fix on the 15.11 branch in order to get Algolia search back up on the 15.11 release. Without this, the 15.11 site is trying to fetch Algolia assets from the main site, and these are no longer there.

Screenshots, screen recordings, or links to review app

n/a

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

1. Configure a local GitLab Docs environment: https://gitlab.com/gitlab-org/gitlab-docs/-/blob/main/doc/setup.md.
2. Check out this branch
3. Follow the steps 2-7 here: !3798 (merged)

Merge request acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this merge request.