For a more primitive type of manipulation, have you thought of face-paint? It should bypass whatever allergies you have that prevent the wearing of masks or other headgear. It won't make people unable to recognize each other, but will interfere in their ability to recognize you - and it may be more ethical to tweak what people's eyes are seeing rather than than messing with people's minds by the hundreds. Because meddling with people's minds like that seems, invasive, and creepy, and inappropriate to me.
So, there are three basic choices I can see in face-paint. You can go for an overall palette swap, tinting visible skin an varying shades of one color - or even a few, if you're willing to follow anatomical contours and be consistent about it - the basic idea being that you are "naturally" that color, so people won't be looking for or recognizing a human-colored version of those facial features. Or else you can go with the exceedingly subtle, which takes some studying and practice and a steady hand, and use the makeup to manipulate the eyes watching into seeing a slightly different face - through suggestions of light and shadow, slight thickening or thinning lines, shading and blending just a bit. The third option, less recommended, is the simple "I am wearing face-paint" option, where all that's needed is sufficient coverage - though this has downsides like being really obvious about hiding identity, and encouraging people to target or track the paint to find who's beneath it, and so on.
The key to this attempt is consistency, of course, you need to convince people not only is that your actual face, but also that you wouldn't use makeup to hide it - so people won't be trying to smear the makeup, or wash it away, or be looking suspiciously at anyone who has makeup kits anywhere they go. The paint itself should be very subtle, thin and easy to wear since it needs not to smear (since all that's needed is a suggestion to misdirect the eye). There should be some paints or dyes which could be found or made that will not smear easily, or will only come off with the proper solvents, that can be comfortably worn long term - thick layers and moist paint (grease based) will be uncomfortable, but something light and dry will be wearable.
Depending on how messy the fighting is likely to be, it might be wiser to tint or shape the civilian face, instead - as being less likely to get smeared or washed off in battle. The tinting would have to be subtler - but even a few shades of tan in the human-normal range might be enough to fool someone's perception, and the subtle facial feature shading should work either way...or even both ways, so an unexpected bare-face might reveal some vanity and the use of make-up, but maybe so not the other identity without the face-paint tricks to get to that face from bare. Shade the nose a little thinner, the brows a bit more arched this way, and rounder nose and thicker brows that way - and not only are your two faces further apart than makeup can do (since it really only needs to do half-way), but people would only ever be comparing your bare face to the face they expected to see, not any other faces that look kinda similar unless they've seen both, and bare-faced-from-both-directions, and also recently enough they can remember clearly to compare side by side.