from the website which so many of you have quoted above, a few things are learned:

1. You cannot overcharge a modern Li-Ion battery. They are designed to stop charging once full capacity is reached.

2. Discharging a Li-Ion battery too much will kill the battery. However, there is protection circuitry installed to prevent that as well.

3. Charging a Li-Ion battery more frequently is better than charging it less frequently, as can be seen by looking at the battery lifetime / charge count chart. The battery capacity is shortened if you wait too long to charge it.

4. Li-Ion batteries do better in cooler ambient temperatures as it prevents nucleation bubbles.

My suggestion, based on the facts is to use the battery that came with your device, and keep it plugged in. There is probably a circuit which is broken when you disconnect the battery, so making a fakie will likely fix that, but then you'd have the device trying to charge the fakie all the time. You wouldn't want that! If it's inside, hooked to the mains, in AC you will be fine.

If you are good at making electronics, your best bet would be to remove the battery, disconnect the charging port, and wire up some leads to the battery terminal which will give you the proper voltage and amps ( 3.7v / 150mAh ).