I have been doing a little investigating of the Teensy Audio Library for a FFT so that I could just extract the phase and bin magnitude (bin size 43 Hz) with the 1024 bin library. Sadly magnitude but no phase. I could roll my own with a Goertzel algorithm but that would involve work.
I am finding that getting a very narrow filter is easy with the 16 Biquad cascaded filters by just dialing up the "Q" parameter. There is a warning to use caution above 0.707 so that signals don't exceed 1.0. Going up to 10 didn't seem to cause distortion problems though it did ring like a bell. I wonder if that could be a good long as long as I can distinguish the Doppler from just ringing.
There must be a million ways of doing this. Any other ideas?