Bill of Materials is really the way to go on this one. The components being invoiced (>? owned parts), assembly charges to track as well as the sale. Setup components as drop shipped to assembler on the PO's or ficticious warehouse overseas to account for the material at assemblers, and finished product inventoried with one production entry. Sell them Bill of Materials. Why? Because from what I have read, they need it and are ready for it.. The business need is there and BOM offers the improvements: to process flow, on accountability, in tracking and in financial reporting. It sounds like the business is growing and ready to move to the next level. Or to put it shortly (cuz I don't know how) see Mr. Nottoli's comment.
Also worth mentioning-
I don't know the details but there are some businesses that obtain some financial 'tax' relief when product is assembled or has some processing done overseas. We use to do this manufacturing ICs by tracking product through Singapore assembly. In this situation material tracking and reporting is critical. If this applies to your customer - how are they tracking the goods now compared to how they could be with BM (or BM & WO).