I think the biggest obstacle would be from your perspective it would just be a steady annoying sound. TIEs only have their characteristic howl when doing a flyby. But if you’re aware of that and you one want it for how it sounds as you drive by, I’d think just playing the tone through the speakers would do.
Yeah, it’s the Doppler effect that creates those cool sounds from the bystanders perspective.
Which, of course, would make listening to that ion engine drone worthwhile.
There’s an audio illusion that’s somewhat analogous to the barber pole illusion — instead of a pattern which appears to always go up or down, you can have a sound which seems to always go up or down in pitch: https://en.m.wikipedia.org/wiki/Shepard_tone
I personally find continuous impossible pitch shifting more distracting than a constant howl, but YMMV.