All parts are here, I can build the charger circuit.

The first step was to build the charger - without load sharing - on a breadboard. These are the parts I used:

• An old Siemens S65 phone battery: Li-Ion, 700 mAh. It has a third pin in addition to + and -, which is connected to an internal NTC (I think) to ground (I think). It can be used to measure the battery's temperature.
• an MCP73832-2ACI-OT battery charger chip and capacitors for input and output,
• a low-current LED with a 1k resistor to indicate the charging status,
• a 10k resistor to set the charging current to 100 mA,
• a 10k pull-up between the battery's thermistor and 5V. Together, they are a voltage divider that is used to generate an input signal for an ADC.

An AVR is measuring both the battery voltage and the thermistor divider voltage and outputs them over USB serial, to be logged on my computer. The battery is currently charging and I'll post a plot of that later if it doesn't catch fire until then. To make sure that the battery temperature rises during charging, I've placed it in my computer's warm exhaust air.