I would like to try some pure DSP approaches and eliminate noise error from old fashioned analog zero crossing detection.  I looked at the Teensy DSP library and unfortunately it only has a magnitude only output FFT and the bins are too big.  I don't have time to to tinker with other FFT libraries for now.  A simpler way is the DFT.  There is a simpler form of that known as the Goertzel algorithm that computes a single DFT bin.   I recall it used for tone detection applications like DTMF decoding with low power micro-controllers.  It offers a phase output as well as magnitude so it should work well and allow "dialing in" the the bin width.  Both a magnitude and phase measurement for the Doppler tone and just the magnitude of the Doppler second harmonic need to be computed.  This should be computationally very efficient though not an issue with the Teensy 3.2 - plenty of CPU cycles.