Li-ion battery?
Can we be sure that "battery servicing" (cycling) has been carried out "as per the manual"?
But maybe the root of the problem isn't the battery at ... perhaps the mechanism draws "too much" current, for some reason.
How do they get on when plugged in to the mains?
I'm also wondering if the "battery empty" is actually derived from a voltage level across the battery ... perhaps it comes from some sort of "clever" circuitry (like a bit of current pulsed through a resistor [LOL] in an attempt to make a guess at "battery capacity"). Who knows.
Anyway, it's obviously an interesting problem - and one that needs to be addressed - so be sure to keep us updated.
