One is not looking for the current with maximum Vz. The Voltage should go up with current over the full range - unless the temperature rise is to much. This point is not what one is looking for. The interesting point is the current at which the voltage is not temperature dependent in linear approximation for the right temperature range, e.g. with 30 C environment or a higher temperature like 50-60C when used with temperature stabilization.
So one would really need to resolve small changes (like 10 µV) in the voltage. So the test rig would be a little more complicated. More like using a commercial DMM (or high resolution ADC) to measure the voltage. The µC internal ADC is more like good enough to read the temperature from the other diode.
If you really want to go with the µC internal (10 Bit) ADC one might be able to only measure the difference to a second reference and use the ADC only for a maybe +-50-100 mV.
So the procedure could be:
0) at RT measure the forward voltage of the second diode, to calibrate it as a thermometer.
1) set test current for a first try
2) Adjust the second ref. voltage so that the difference to the DUT is small (e.g. < 50 mV)
3) Put a relatively high current (e.g. 10-20 mA) through the second diode in the chip, to cause some heat up
4) Change the current to the second diode to a small one in forward direction to use it for temperature measurement
5) do a fast measurement of voltage difference and temperature during cooling (e.g. 5-30 seconds). Temperature measurement is not that critical - decay curve should be similar for the devices.
It more to get the average temperature and an estimate for the order of magnitude.
6) calculate / estimate TC for the given test current and temperature range
7) adjust the current to get lower TC (e.g. interpolate / extrapolate form older points and maybe typical curve).
Repeat if needed (TC to large).
For just a few parts one could do this also by hand - reading the drop on a DMM.
If you really need to automate, I would look for a better external ADC (e.g. 16-24 Bits).