The advantage of having two of these is that if something's different, then something's wrong.
In this case, one of my units (the one that is still pristine with no serial port) will not hold a charge (with the power switch off) once the charger is removed. The other is just fine and the battery has stayed solidly at 4V for 24 hours. This is also the one that arrived with a completely discharged battery, whereas the other one worked out of the box.
I charged the offending unit until the green LED came on, then took it off charge. Here are the battery voltage readings over time:
11:30AM: 4.0V
5:47PM: 3.7V
11:00PM: 3.6V
6:50AM 1.0V !! (that is, this morning it was essentially dead)
I've swapped the batteries and the problem stays with the unit, so it's not the battery. During development, Chris, that you said you were having a similar problem, but then had found a solution; I don't know what that was since you never posted the 2nd schematic, but perhaps this one doesn't have that fix?