... see whether the noise was caused by the sample data or an artifact of the Arduino output process?
Indeed, that's a possible explanation. In the OP's wav file, I hear background tones that shift by an interval of about a fourth, with a shift frequency of a one-second order of magnitude, while the froreground tone sounds to be almost constant. I don't readily think of a process that generates such an orderly kind of noise. I'll admit, though, that I can't do Fourier analysis like that in my head. There may be any number of ways that the background tones could appear, that are beyond me just now.
I'm content to wait until the inductors are delivered, and see what the five-pole Chebyshev filter yields.