Backup, Mirror and Snapshot will always contain all the data in the database and won't let you filter just the data of one customer.

Moreover, snapshots can only reside on the same server as the source database, so they're out of question here.

What you can reasonably do is export the data with an ETL (SSIS to name one) using some custom logic.

Another possibility is a merge publication, using the customer as partition. The downside to this approach is the increased load on the publisher database due to merge replication triggers.

Personally, I would go for a custom SSIS package.