With temp change a thermocouple is a varying DC voltage.
Thinking on two connection tip.
That has to effect the circuit. You should see a DC offset in the AC. The problem is that it's a small signal and needs to be made large to get a reading.
So if you have one path that has a thermocouple & a second path that does not have a thermocouple and you power both paths with the same signal ( the power ) you should be able to see a difference.
You have wires in path that change but should effect both half's or AC the same.
You have the heater resistance that should change based on heater temperature but again should effect both half's of AC the same.
If you put a current sense resistor in both paths and measure the difference you should see the voltage of the thermocouple voltage. You would be building a Wheatstone bridge.
https://en.wikipedia.org/wiki/Wheatstone_bridgeIf Rx = cable to handle & tip. Then with proper adjustment of R2
you should get a AC balance with thermocouple effect showing.
Sensing the heat of current sense resistor should give an idea of heater temp.
By taking a lot of samples, you should get a lot of information.
Rate of thermocouple helps build profile of tip & gives hint of mass tip is in contact with.
Would not be that hard to build some test plots or graphs to see what you can get from data.
Just need a few differential input ADC's that sample at same time to remove time errors.
Starting to wonder if they could be using some audio ADC's & DAC in a circuit.