Phase Modulation and Intermodulation Anomaly ? (Spectrum plots inside - **Updated**)
A couple of points to think about:

1) Looking at the scales of your FFT plots, it appears that the fundamental is around -25 dBW, whereas the 5th harmonic is around -125 dBW. This is a very large difference, implying the system is only weakly nonlinear. In practice you'd have problems detecting the 5th harmonic even without the phase modification just due to thermal noise. If you change your scales to have a dynamic range of say 100 dB, you would see the two plots look nearly identical.

2) The phenomenon you are modelling looks to be AM-PM conversion. Under this interpretation, it can be thought of as "adding" some phase noise to the output signal. (The tails of the phase noise spectrum would droop down and mask weaker signals.) This could be an explanation for why the result looks like a noise floor. 

3) A sanity check could be-- what does the spectrum look like when the peak phase shift is halved? Or cut to 1/4th, etc? Does the noise floor drop by some commensurate amount?