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The reason for microphonic sensitivity could be either a piezoelectric effect, or (maybe) piezoresistance. Piezoresistance (actually leakage in the insulator) would have response at twice the injected frequency.
Small-size ceramic capacitors use piezoelectric materials (rutile and titanates), but a wide range of ceramic materials is available, you'll have to buy and try. NPO should be fine, but C values are small.
Polymers aren't intrinsically piezoelectric; plastic film becomes polarizable because of internal strains, so look for mechanically weak dielectrics. I'd expect polystyrene capacitors (like Mallory SX series) to be nonmicrophonic just because styrene is a weak plastic.
Aluminum electrolytic should be fine for your low-pass application. That dielectric, Al2O3, is definitely NOT piezoelectric.