Hi Blood,
Here is what I would do:
Modify the wiring around the thermistor (but leave the thermistor installed in the iron) to add a 100kΩ pot wired as a rheostat (only two wires) in SERIES with the thermistor. Initially set the pot to zeroΩ and the iron's built-in control to minimum temperature. Heat up the device and let it stabilize. Now turn the new pot to add resistance in-series with the thermistor, say about 10KΩ at a time. If the temperature drops, then keep increasing the resistance until the new temperature decreases to what you want. If the temperature increases as the added pot is increased, then go to plan B.
PlanB. Rewire the added pot in PARALLEL with the thermistor. Initially set the pot to maxΩ (>100KΩ) and the iron's built-in pot to minimum temperature. Heat up the device and let it stabilize. Now turn the pot to reduce the effective resistance of the thermistor. If the temperature drops, then keep decreasing the added pot resistance until the new temperature is what you want.
If either of these gets you the correct new minimum temperature, then remove the added pot, measure its final value, and add a fixed resistor of that value either in series or in parallel with the thermistor, which ever worked. The built-in control should still adjust the temperature, just from a new, lower starting point. If I guessed wrong on the thermistor resistance, you might have to repeat the tests with a 10kΩ added pot instead of a 100kΩ one.
Keep one hand in your pocket while working on this. The circuitry might be connected to the AC line with no isolation.