Separate key-related frequencies from audio signal
Attenuating the harmonics of your instruments isn't really a viable option, because you'll end up annihilating the signal itself. The harmonics here are what are sampling the behaviour of your instrument.

I'm not sure if I'd call it simple, but a possible solution to separating the contribution of pitch in such a classification system might be cepstral liftering.

This is, calculate a cepstral representation of the recording of each instrument (the Fourier transform of the logarithm of the frequency response) and separate the periodic components (pitch) from the aperiodic (perhaps timbre?). Such separation is known as liftering - the analog of filtering in the cepstral domain.

The separated signals should roughly correspond to a periodic signal and an impulse response better characterising the instrument.

This is an approach used in to separate the harmonics of speech (carrying pitch information) from the transfer function of the vocal tract (which generates a filter forming vowels amongst other characteristics).

I'm not sure if this will work in your case, but I think it would be worth exploring.