Bluesky: IMO the only "easy" part to your question is "how whould infringement under the DOE be judged?" IMO the anwer is simply, based on the (in)substantiality of differences. Not very satisfying or particularly helpful. The FWR "test" for substantiality of differences is most helpful in a mechanical case. Does the screw in the accused device perform substantially the same function in substantially the same way to achieve the same result as the nail recited in the claim? Etc. What does "substantially" mean anyway? FWR can be helpful in some chemical cases, e.g. substituting Mn for Mg in a welding flux. By their nature, eqitable doctrines are "flexible". Geat if UR drafting/prosecuting, not so great if UR trying to draft a FTO opinion (DOE drives me nuts sometimes).
Your analysis may depend on the technology area. If A = temp and AA = pressure, IMO on the face substantailly different. But I can envision that, in some cases, T & P might just be surrogates for each other (i.e. heating a closed system of constant volume). In other technologies, I'm thinking computer implemented inventions, a surrogate relationship may not be so easy to find and your DOE argument may well hold water. One has to look at the "whole ball of wax". How/when did that limitation get introduced into the claim? What did applicant say about that limitation during prosecution? Is there some "cut-off" to the DOE in your case, e.g. "disclosed but not claimed", would literal substitution of AA for A "render the resulting (hypothetical) claim anticipated or obvious over prior art", etc. Good luck!