If the packaging bears the European buyer's brand, then the person who "commissioned the design or manufacturing and who markets it under their own trademark" is the buyer, and the buyer may be the manufacturer as defined by the regulation.
This is actually good news for the Taiwan supply side—the documentation responsibility for this line is not on you.
But be sure to clarify this in writing first. If both sides assume the other will handle it, you will have a problem on the day of shipment.