Nonlinear effects of grille cloth

It can't really get much closer in this case. The cloth is stretched over a thin hoop made of white oak (steam bent) which fits snugly over the driver's front gasket.
Have you made sure that it is not actually the grill 'frame' vibrating or being affected that causes it?
Grill frames as mentioned before is a common source of diffraction, and 3rd harmonic from rattling/vibration etc.

Without trying with fabric only and check the difference, the measurements can't give a conclusion.
Did not see it mentioned by anyone else the Weltersys.
 
It would still be interesting to see the Fourier transforms for a sampling of frequencies.
You'll have to be more specific as to what kind of analysis you had in mind. To me, a "Fourier transform" with no other qualifiers just means the frequency-domain representation of the impulse response.

Have you made sure that it is not actually the grill 'frame' vibrating or being affected that causes it?
I'm fairly certain that the hoop is not the culprit. The other driver I measured was in a separate cabinet with a separate grille assembly (which happens to fit slightly looser), yet the results were virtually identical. No grille cloth:
no_grille_2.png
With grille cloth:
grille_2.png

The relative microphone position is not identical to the plots in post #1, which is why the fundamental magnitude differs a bit.

How fast does the distortion rise with SPL?
Does supporting the grill cloth have an effect? If the cloth experiences nonlinear air resistance, then it becomes a radiating surface for odd harmonics.
I'll try to do a few more tests soon. Thinking about it, it seems plausible that a rigid support may have some effect given what looks to me like some degree of frequency dependence. If one assumes a rigid material, the nonlinear contribution should depend only on the sound pressure.
 
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I finally got around to doing a few more tests. The microphone position is roughly the same, but a few other things make these measurements not directly comparable to the previous ones:
  • There's a passive filter network between the amp and woofer. I filtered the test signals to keep the voltage at the woofer terminals approximately constant up to ~2kHz, but the woofer's distortion profile is still affected by the higher driving impedance.
  • I put a piece of reticulated foam in the pole piece vent. I initially thought that this might help reduce the distortion spike around 300Hz (which seems to be related to the vent and the cavity behind the dust cap), but it mostly just increased the LF distortion a bit.
  • I used multiple sweeps to suppress noise.
  • I used a different USB audio interface (probably not much effect).

First, no grille at approx. the same drive level as before (~6.3V RMS at the woofer terminals; ~5W into 8 ohms):
no_grille_5W.png
And with the grille:
grille_5W.png
Next, I tried to support the grille cloth somewhat by holding a couple hardwood dowels across its face:
grille_support_5W.png
I couldn't tell from where I was standing during the previous tests, but it turns out that the grille cloth does move quite a lot. Tentative conclusion: The motion of the cloth is a factor, at least at lower frequencies.

I also tested two lower drive levels. Here's ~2.83V RMS (~1W into 8 ohms) without the grille:
no_grille_1W.png
And with the grille:
grille_1W.png
Still some difference, but smaller than at 6.3V.
Now ~0.89V RMS (~0.1W into 8 ohms) without the grille:
no_grille_0.1W.png
And with the grille:
grille_0.1W.png
Seems to be no clear difference at this level. Note that 0.1W is still moderately loud in the context of home music listening—around 88dB mid-band at 1 meter in half space.