Never tried this but…
- Run the clone link of your project, creating a clone copy which will include the old table.
- Rename the cloned table for clarity.
- In your original project “create a new table” with the + symbol but choose to use the database linked from the “clone” instead of creating a new one. (I don’t use the beta editor, so I’m not sure if this is available in it yet?)
- Delete your cloned project and it should leave your linked clone table with the original project as the only “owner”
Definitely not “one click” but it is better than a csv and re-upload.