Is it ever alright to add a subtitle to a title shortened by publication when citing? Let's assume that a paper was submitted under the "Extremely long and necessary title that explains you are studying pieces of Class A, Class B and Class C of this Bigger (and much more frequently searched) Topic". 

This, of course, had to be shortened for publication and now is published and cited under the title:

"Extremely long and necessary title that explains you are studying pieces of Class A, Class B and Class C"

Still correct but likely to fly under the radar of many researchers. So, is it ever OK in a citation (APA or in general) to add the [additional subtitles] when displaying the citation like below? Is it even ethical to adjust the title post publication?

"Extremely long and necessary title that explains you are studying pieces of Class A, Class B and Class C [of this Bigger (and much more frequently searched) Topic]"

I'm a new academic and I appreciate any thoughts.
I'd stick with the published title.  You definitely shouldn't edit it without indicating that you are doing so, for example with square brackets, and I'd avoid that too.  I've never seen anyone do it.  If you do edit the title, then the best case scenario is that you'll come across as a little eccentric, and you may confuse or irritate readers.  (They may wonder whether you are referring to a slightly different version of the paper than the one they have, for example a longer preprint or working paper that was shortened for publication.  Then they may waste time looking for it, or decide you are being sloppy about your citations.)  It may also throw off citation counts, which is not in your interests.  Incidentally, I assume you are talking about a previous paper of yours - you certainly shouldn't edit someone else's title.

Instead of editing the title, you can convey this information in the body of your paper via your citation.  For example, you could write "For background on Classes A, B and C and how they fit into Bigger Topic, see Paper X."  If you give the reader enough information, then the shortened title really won't matter.