Somewhere on the board will be two resistors which go from the two terminals on the battery, those two resistors are the business components of the battery indicator circuit, it is possible that the tolerances of these resistors are making your low battery light switch on prematurely.
Just measure them with a multimeter, then compare to the value written on them.
If its way out, that is the cause of your problem.
Chances are they are low tolerance components anyway.