Assuming they are lightbox/EL panel (or similar) flats, they would not do anything to solve a gradient caused by an external source such as light pollution (whether natural, i.e. moon, or artificial). They will only deal with uneven illumination caused by the imaging set-up, i.e. vignetting, dust shadows and differing sensitivities between sensor elements. Not that you shouldn't use flats of course, but they aren't going to deal with this problem.
Background neutralisation is not going to deal properly with the gradient either as you have found, and should be applied after you have eliminated the gradient, not before.
You're on the right track with DBE. You could apply it after you have assembled the final stack of images, but since the problem is isolated to the Siii images you may find you get better results if you apply it to the stacked Siii before final assembly with the other filter bands. I don't know if each of the individual Siii frames has a different gradient if the moon or your framing has changed significantly as the session progressed, and if so there might even be merit in applying DBE to the individual subs before they are stacked, but given the interactive nature of DBE it might be too much effort to justify unless the object framing is consistent between frames.
I guess you might find that you get an odd 'colour' cast by applying DBE to only one of the filter bands, but since the choice of colour balance in a non-RGB image is arbitrary it isn't a major issue. You can of course use Background Neutralisation (or any other method you prefer) to get rid of unwanted colour casts in the background.
