Have you considered a positive feedback capacitor with the op-amp, to try to make an oscillator?

Then the illumination on your LED will affect the phase of the oscillator. By trying to make it oscillate at ~10Mhz, and then sampling the data, you should avoid the clipping problem, and you'd have a lot more data to use to potentially extract interesting things with clever math.

Even if the oscillator runs at 10Mhz, you don't have to sample at 10 Mhz - you could sample at a lower frequency (which must be clock cycle accurate) and make use of aliasing to still measure phase shifts very accurately.

You're totally going to measure the power supply, temperature, radio signals and a whole host of other things at the same time, but with the right signal processing/modulation you should be able to extract the effects you need.