Make sure to get an oscilloscope setup that is able to detect high frequencies in good shape, as CFL's can be driven with hundreds of kHz. An engineer might be able to refine the setup with a transimpedance amplifier circuit. If he succeeds, perhaps he could contribute his knowledge to the oscilloscope thread to help us increase the accuracy: https://ledstrain.org/d/312-homemade-oscilloscope-to-detect-pwm-diy-guide/79
I found that CFLs are better for me spectrum-wise than regular White LEDs, however they usually do flicker a lot (100/120 Hz plus 20 kHz onwards, on top), causing flicker symptoms.
I think a major factor for eye strain can indeed be spectrum. I am very sensitive to White LEDs (the technology "White LED", not some "warm" or "cold" spec on the package) and have to turn the brightness all the way down. The few displays I'm comfortable with at high brightness have something in common: a spectrum consisting of narrow lines as opposed to having a rainbow spectrum. AMOLED and Quantum Dot. Of course, such displays become unusable when there's pixel flicker or backlight flicker going on, making it very hard to isolate spectrum as the single cause.
If I look at a CFL's spectrum I can see distinct lines there as well.
It would be interesting to find out how it is possible that spectrum has such a huge impact.