The system needs at least 10k of resistance across these two pins to power on. I assume that this safety feature forces a power-off if the battery gets dangerously hot?

Usually, it is to protect against the battery exceeding a safe temperature when charging; however your system may cause the battery to be completely disconnected in an attempt to eliminate any current flow in or out of the battery. Stopping current flow in or out helps to allow the battery cool off.

You really need to look up the specifications for your particular battery that you are replacing the original with, and determine the maximum safe operating temperature, normally given on datasheets. If you can't find a datasheet, I'll suggest that exceeding 140°F will be unsafe. It would be better to provide a margin for safety and stay well below that.

Then you need to determine at what resistance the device charge/discharge circuit disconnects/turns off at.

Then you need to find a thermistor that will cause the circuit to stop charging/discharging when the temperature gets too close to 140°. The thermistor must be firmly in contact with the battery pack to assure a good heat transfer, preferably between a couple of the cells at the middle of the pack.

Your OEM battery pack may have a thermistor that forces a low rate of charging due to the battery pack being "buried" inside, and/or concerns about the possibility of burning/scalding humans that come in contact with it.

We have to be concerned about safety here. It would be negligent to overlook the consequences of overriding the safety features built in to your device, the defeat of which could very likely result in disaster.

I'll suggest one way that you can test the charging circuit; connect a 10 Ohm 10 Watt power resistor across the terminals of your device without the battery connected, and then use a 20k to 50k pot wired as a rheostat with a 100 Ohm resistor in series with it instead of your 10k fixed resistor. Then measure the voltage across the 10 Ohm resistor as you change the resistance on the pot slowly.

Find the point where the charging starts, and measure the pot + 100 Ohm resistors' total resistance.
Then find the point where the charging stops, and measure the pot/100Ohm resistor again.

That will give you a starting place for finding a thermistor with the correct range.

 

If I did this mod, I would not use the internal charger anymore. I would just wire the battery terminals on the device to a quick disconnect. This way, I could disconnect the LiPo (Lithium Polymer battery) that I have modded in and charge it with my LiPo charger.

Check out the third picture in post 12. The square, red plastic piece connected to the wires is what you guys would refer to as a "quick disconnect." We RC pilots call it a Dean's plug.

Hope that clears some things up...