The 7xx built in charger works great. Never had an issue with it, almost never used my external charger. I question whether there is such a thing as a "smart charger" for the lithium camcorder batteries. They aren't like nimh batteries or nicads in that regard. They don't have memory. They have protection and charging circuits in front of them that limit how quickly they can be charged and how much load the cells see.
Another critical point... Like all batteries, these batteries are subject to drop damage. The fact that they are lithium means the consequences can be more serious. If a cell is compromised, bad stuff could happen.
I think it is essential to monitor the voltage from your battery on the recorder... Note it before you start, note if it is abnormally low. At set break, is it where it usually is? Maybe it starts at 8.0 or 8.2 and at set break it is 7.6, and 7.4 at show end... Ideally, we would have a graph with time vs. voltage. That would tell us when a problem was developing. It would be a great 7xx feature. It would also be possible for them to track the rate of voltage discharge and issue a warning to the user. Sort of a "battery suspect" warning.
Lithium batteries do not like heat. It reduces their effeciency and it shortens the life of the cell. It is possible that the battery has a thermal shutdown.
I had great success with the 6000 lithium in my 722.. I ran the same battery night after night and charged it in the unit, almost never removing it.