Answer
Add a read-only custom field to the Sales Quotation Lines screen that pulls the Business Opportunity Number from the underlying view — since Window Transfer relies on an actual field being present on the source window to carry the key value across, surfacing that otherwise-hidden value as a custom field first gives the Window Transfer something concrete to reference. With that custom field in place, configuring the Window Transfer against it (rather than trying to transfer using a value that was never displayed on the window at all) correctly opens the Business Opportunity window populated with the right record instead of blank.