I doubt it can work. Noise cancelling in earphones works good only because it is very narrow ear channel and you are not dealing with any phase shifts due to changing position relative to noise source. Also there is no other things like reverberation and room acoustic which also contributes to phase of the signal that we actually recieve in our ear channel. In other words noise cancellation is possible only in one specific position of listener in the room.
Yup. I'm still learning- but it seems like a Helmholtz resonator would be best in this case.
Since a room has reflections and diffraction of the sound waves, the 312hz they're trying to dampen w/ ANC will be at different phases at different parts of the room. It's also not a point source.
I would propose funny solution to change the room where 312hz will fall into one of the standing wave frequency. Then if you combine properly where to place fan and where to place your sit, you can achieve effect of cancelling at this frequency/position (and amplifying in other positions)