Draft: Make sure Docker images are pushed when testing

What does this MR do and why?

By pushing the Docker images in a test registry, we're effectively emulating exactly how the production images should behave.

After this is merged, we should set up clean up policies to delete the test images.

See #1720

Container registry of test images: https://gitlab.com/gitlab-org/gitlab-docs/container_registry/5618636

Merge request acceptance checklist

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