Sorry for the delay in replying. I wanted to try everything before reporting back again. Interestingly, of all the procedures I used, only one did the trick. I tried various cleaning methods as recommended above, and only Goo Gone made even a temporary difference - but the stain came back as it dried. Before I started the last trial, the case looked just as bad as it did when I received it.
So I poured a thin layer of generic vegetable oil on the area, and used a cotton swab to make sure every area had some oil on it. Fortunately, the layer was thin enough that with the case level, it didn't run off or spread beyond the area I wanted. I let it sit for almost an entire day and then started wiping it off, rubbing fairly hard with a paper towel. A very small amount of dark stain came off on the towel, but the case stayed its proper color as you can see! I did not use any cleaner afterward, because wiping the oil off seemed to do an adequate job and the plastic did not feel moist to the touch. In fact, it had the shiny and smooth/satiny feel of the rest of the case.
After the treatment, I've left it sitting for three days before taking the picture. There is no sign of recurrence. My opinion is that the stain was actually an area where the plasticizer was leached out of the case material, and the oil restored enough of it that the plastic resumed its original appearance. After doing some research online, it seems that vegetable oils are used for plasticizers in some substances to reduce the use of toxic substances, so there is some plausibility here. I may try another treatment as the original coat of oil gets absorbed into the plastic, because as it soaks up the oil the surface may dry out again. We'll see.