Fan noise is painfully irritating in part because it’s constant and relatively spectrally pure — it’s not a sinusoid, but it’s almost all harmonics of a particular frequency, so the sonic energy is concentrated with high Q in a small fraction of all possible frequencies.
In experimenting with bytebeat I’ve found that a bit of envelope modulation at 0.5–2 Hz can go a long way toward reducing the pain level of tones. Could you do that with a fan? For example, closing the fan with a baffle for 500 ms every 2 seconds would nearly silence it.