Once you are in a specific context within edit-site (or a context has been loaded) there should be a way to create a variation for that specific context (if it's singular, a singular-{post-id}), if it's a monthly view of archive a date template and so on. This flow is decidedly more advanced, so it should be handled carefully and be easy to disable, but it should be possible.

These templates would be saved in the CPT for the site's wp_templates directly.