I think this post is going to come across as stubborn and needlessly probing, but that's really only because I'm still a bit confused as to what exactly is wrong with what I'm trying to do. I do understand that what I'm hoping exists probably just doesn't, but I still feel like I could gain some additional insight into how close I can expect to come.
Treat the stored value as a maximum and modulate it based on view direction, but even then the modulation requires some light direction to compare with the view direction, or else you're back at a Lambertian model.
(...)
In summary: To model a change in reflectance as viewing angle changes, an incident light direction is required to compare to the viewing direction.
I'm not really sure why this is (and what do you mean by "or else you're back at a Lambertian model"). It seems to me that a Lambertian, ideally-diffuse surface implicitly assumes that a) the apparent luminance is the same from all viewing angles (that is, light leaves the surface according to the law of cosine with the viewing angle) and b) that this luminance is determined according to the law of cosine with the light angle. There's not really anything explicitly preventing me from violating a) without taking into account the light angle.
The assertion that "no such BRDF exists" is actually kind of trivial to refute: like I said before, I can make a BRDF that just assumes there is some kind of rim light in the background that moves with the camera. It wouldn't be physically plausible, and the result wouldn't vary based on all of the information input into the BRDF (but so what? A constant function is still a function), but it would still be a BRDF. My question, I guess, is why am I doomed to, at best, "try and fake" what I'm looking for (even just successfully faking it is all I'm hoping for)?
On that note, can you elaborate on what you mean by "has more in common with the phong model specular term than its diffuse one"? The Wikipedia page on Oren-Nayar has a table at the end that contrasts it with Torrance-Sparrow specular which seems to suggest otherwise, and, like I mentioned, it doesn't "look specular," (which I know is a stupid thing to say.)
The reason I was inspired to ask this question in the first place was that when I was playing with the Oren-Nayar shader in Blender, I initially assumed it didn't take into account the viewing angle at all -- that is, that light left the surface according to the law of cosines, but that the amount of light that left obeyed a non-Lambertian law (which presumably makes no more sense that what I'm hoping to find). I fairly quickly realized that that wasn't the case (and I also realized that it was a lot more similar to Blender's Minnaert shader than I had thought, which I'd in turn thought was the elusive dependent-on-viewing-angle-but-not-"specular" thing that I'm now seeking and probably doesn't really exist), which lead to my confusion: if it does depend on both the viewing angle and the light angle (which, as you suggest, makes it more like a non-mirror "specular" term), why is it almost invariably referred to as "diffuse" in contrast to "rough specular."