Hi,
Using the translations - you will have to ensure the documents have the same IDs and names or, edit the XLIFF files to match the other site document data.
The obstacle here is, I guess, the culture of the documents - the culture code is one of the identifiers of the document - so, staging or export/import modules won't help between the sites with different cultures.
You will have to modify the translations and the xliff files or create custom tool that will transfer the data using the API.
Best regards,
Juraj Ondrus