Phase response distortion due to IQ demodulation
I’ll hazard a guess on this that David’s comment is spot on.  You could help by providing some specifications for the low pass filter attenuation at the frequencies at which you are observing the phase distortion, or perhaps provide plots of your magnitude/phase responses for the input and output. If this is what you’re observing then:

1). Yes. For a DFT or FFT of a signal, there is likely to be phase distortion at low relative amplitudes based on different implementation details. This has to do with the precision/resolution of the data on your machine.

2). Sort of.  The filter could technically be reversible in the analytical sense (ie on paper) but quantization error from the limited precision of the data will make this impossible with many many applications.

3). Recover?  No, not really.  Assuming that you don’t have any errors in your implementation, and it sounds like you don’t, then that is simply the nature of the beast.  However, you are using filtfilt, which I understand works by applying the filter forwards and backwards, resulting in a net zero phase response.  As such, it would follow that your phase response should be unchanged, which I assume was your original hypothesis. You could just use the phase response from the original signal, if you have it.  This may or may not be practical for your application, and could be done more efficiently if it were an option.