I tried to do an experiment where I changed the distance of the light source from the spectrometer slit to see

a) if the inverse square law holds ;-) b) if there occur any changes in intensity peak wavelengths during overall light intensity changes

I put the spectrometer beside the starting point of a ruler and the light source (a cheap LED torch light) on top of a little box that can be pushed along the ruler to measure distance between light source and spectrometer. I took spectra in 5 cm steps (about 2 inches) from 80 cm to 30 cm.

Here are the first resulting curves overlayed:

https://spectralworkbench.org/sets/3252

I would not put too much trust in these data because there was a "light source limits problem" which I mentioned in an according research note. I will have to do the test again with a better light source and a more stable setup to achieve the goals of this experiment.