The current mirror could come in handy because it avoids putting a resistor inline with the component you're measuring. It's a way to use Easy's recommendation:
"If you convert current to stiff voltage and apply that, ditto.'
You can take voltage-vs-current readings through a transistor diode junction.
Since you speak of a power supply 8-28 V, you can make a resistive-divider divide down what goes through the left transistor, so that you measure its current, whereas the right-hand transistor gets tested in its turn-On range of a few tenths of a V.
You can either a) apply voltage in tiny increments and measure resulting mA,
or
b) apply mA in increments, and measure what voltage is able to create each reading.
I once did similar tests on a real diode to discover what data is generated. I measured at different values. I had to carefully adjust my variable power supply, painstakingly. I had to read my meter, painstakingly. I plotted the results on a graph. My research took me a few evenings.
Good luck.