Delbourt

Here is a graph showing resistance change with temperature for a thermistor.
Presumably the device in the car will have an initial value, and once hot that will or should change.
It might be worth getting a connector (jtpo?) fitted with two wires that could be connected to a multimeter measuring resistance then starting the car (if you can with the easy start) and watching what happens to the resistance as the engine warm up.
Risk is all yours as in the usual way.
Potentially an alternative might be to measure resistance cold, then reconnect the standard wiring. Start the engine and let it warm up to standard conditions, then stop the engine disconnect the wiring and measure the resistance. Couple it all back up and restart and then stop and cool down naturally.
Otherwise change the transmitter and see if it cures the problem.