I believe my watchmaker said the balance shaft needs replacement on my Longines cal 431, and he thinks the easiest solution is to buy a spare movement for the part(s). PM if you have any leads! I've been trying eBay so far, to no avail... Edit: I just spoke to the technician (WM wasn't in today), and he thinks a movement was recommended because the setting for the balance shaft was drilled out to fit a non-OEM shaft, so it looks as if a full movement is what I need.
@Spike sent me a link to an eBay seller (Peter, in Hungary), who gave me the information I needed to find the specific part. The part has been ordered from the UK. Hopefully, that part (the click to drive the date, #2649) will solve the 'non-running' issue without buying a complete movement!