Just to muddy the waters:
devin, are these sustain/mute "thumps" at all consistent, one piano to another? Whatever the sound signature, I wouldn't think that this particular "noise" is anything like the sounds a piano can make or might be expected to make. And are the microphones picking up the thumps sonically or mechanically connected to the pianos?
I know nothing of the capabilities of a profession mixer. But if these thumps could be
characterized (see 1. below), perhaps they could be
specifically filtered out, either by the mixer or custom, passive,
adjustable filters (as opposed to the momentary, general
reduction/muting of ALL sounds from the piano) .
1. For instance, this paper:
https://digital.library.unt.edu/ark:/67531/metadc30427/. Excerpt: "... The first portion of this paper describes methods for quantifying noise content using FFT analysis procedures. ..."