I suspect it'll be easier and more robust to proxy to the pages daemon with appropriate rewrite rules than to use the Gitlab API.
If you do go via the API you'd need to ensure that every *-site repo uses the same branch name (I think) and ensure that the artifacts for the latest run are saved forever. For instance, it looks like the latest pages job at https://gitlab.common-lisp.net/cl-docker-images/cl-docker-images-site/-/jobs... has already had its artifacts purged.
-Eric