Odd that it is only one filter that does this. I can think of one possibility specifically for NB filters.
When light strikes a NB filter at an angle the wavelength band pass shifts. This is why faster lower F-ratio systems need special NB filters, the lower the F-ratio the greater the angle of the incident light cone on the filter, and the more the bandpass shifts. But also, the farther from the center of the FOV, the greater the angle of incidence away from perpendicular, and the more the shift. So in fast systems with a NB filter, the center of the image is brightest, the farther away from the center to the edges is darkest. All this depends then on how fast your system is, how narrow your filter band pass is (very narrow, definitely a stress), and of course how big your sensor is (full frame, another stress). The other thing is filters have an opacity to other wavelengths, this could change with angle too. And there are reflections from filters, so reflection/scattering might differ as well from center to edge. And finally this depends on the structure of the filter, its reflectivity, refractive index, thickness, and what band pass the filter is targetted to, and of course, exactly where the filter band width centre is positioned with respect to the NB line of interest. So behaviour can be very different with different filters.
All this being said, I would have though a correctly made flat would show the same behaviour as the light, and fix these types of problems. However, a very bright flat illumination can penetrate through the filter depending on it’s optical density and the angle, while the much fainter sky would not. Do note though, the more severe the vignetting, the more difficult it is to get a perfect flat.
Some things to try.
Don’t use any correction, and then see how bad the vignetting is with the different filters, is the OIII vignette much worse compared to Ha or SII without using any corrections for your lights? If so, the worse vignetting may be making your life much more difficult to get a good flat.
Make sure your OIII flat is not clipping or not too low at the bottom or too high at the top of the histogram, you want it to be in a nice linear region in the center of the range.
I don’t see how a bias or dark would affect this, as long as your bias and dark are good. I have had my bias and dark change on me suddenly, so make sure you do new fresh ones.
Personally, I always do biases, darks and flats in otherwise dark room, to prevent any stray light, doing flats in a bright room could result in a light leak somewhere in your system.
Rick