The easiest way to handle these kinds of experiments is to pick a voltage that is 'about right' and measure it with a DMM. Get as close as you can to the desired value. Do the experiment while measuring everything with DMMs, preferably simultaneously. All meters should be calibrated annually - and we KNOW that won't happen! So, all of this supposed accuracy isn't traceable - it is pretty much meaningless but we carry on...
We pick another voltage nearby and measure again. We rinse and repeat until we have enough points to make a graph.
It's a matter of choosing the tool. A high resolution power supply is pricey and unless it is of the 4 lead Kelvin type of connection, the power leads voltage drop won't be included in the readings. A DMM, OTOH, measures right at the load. It doesn't care about voltage drops in the leads. And it can measure voltage and current. Some can do it simultaneously but they too are high $ items. Current measurements change the associated voltage measurement due to voltage drop in the meter's shunt resistor. So, measure the voltage after the current measurement.
High resolution DMMs are cheaper than high resolution PSs. That's good because you need two or three of them.