Before the analogue watch stopped, the LCD module had begun to display numbers that wouldn't change. I reckon the power draw of the LCD is higher than that of the watch movement, so I think in future I will have the battery changed as soon as I see the LCD acting up, rather than waiting for the watch itself to stop.

Jeff, might be worth having the battery changed on your Memphis too, just in case it is on the way out and can't power the LCD properly even though the analogue watch still running makes it look like the battery is fine. Worth a try at least, a $10 fix has got to be easier than trying to source a new module!