Identify fundamental frequencies in short-period waveform
Finding the sum (and/or difference) of source frequencies in an FFT result usually means that there is an inter-modulation distortion in the source mixing (mixer).  Likely a multiplication instead of a strictly linear summing of source signals.

Regarding your last bit of thinking, trying to do blind source separation using just a short FFT is trying to do something unusual.  Any DSP solution is extremely unlikely to be simpler.