Backport EE changes for built-in project template test

What does this MR do?

Backport of https://gitlab.com/gitlab-org/gitlab-ee/merge_requests/15766

Does this MR meet the acceptance criteria?

Conformity

• [-] Changelog entry for user-facing changes, or community contribution. Check the link for other scenarios.
• [-] Documentation created/updated or follow-up review issue created
• Code review guidelines
• [-] Merge request performance guidelines
• Style guides
• [-] Database guides
• [-] Separation of EE specific content