How to use AP Style commas after dates
Solution 1:
Unless your instructor can cite chapter and verse of the AP Stylebook, consider it a rumor. Anyone who can claim total knowledge of a 400 page document with 200+ changes in the last year alone is probably lying or at best mistaken.
Your reasoning is sound, but I don't have access to the latest Stylebook.
Solution 2:
As msw says, it's probably rumor.
I would leave the comma in after 2012 because the sentence
April 4th, 2012 brought the death of my cat.
has an extra bit of white space before the year, and no extra space following it. This optically brings the year closer to the word "brought" than to the word "4th", which goes against all visual sense. (In my mind.)
Additionally, one could read (admittedly I am going out on a long limb here) that on April 4th your cat's demise was brought about by the machinations of the devilish year, 2012.
As one of the Fowlers probably would have put it, this sentence needs recasting. But then again they weren't American newspapermen.