It isn't fair to you. (I am assuming you are telling the truth here, by the way.) But, on the off chance that you are the type of person to cry "sexual harassment" when someone commented on how cute you look today, they don't want you to bring those problems into their office. With the unemployment rate being so high, there are numerous other candidates available, so why take the risk?
So, you've already discovered that sharing too much information doesn't work and that lying may work temporarily but it leaves you wondering what will happen if you get found out. And you may get found out, which could result in you resigning from your current job only to have the new one yanked out from under you when the truth comes out.
So, no lying. But don't stress out too much about it. The reason you are looking to leave your current position after such a short time is far more important than why you left your previous job. You need to be able to explain, in a positive way, why you are looking. The longer you are in your current job, the less important the last one is. You may want to sit on your resume for a while and suffer though this job for at least a year.
Either way (staying put or looking), when asked keep your reason for termination short. On a written application write, "involuntary termination." With all the layoffs going on, most people will just assume you were laid off. (And you certainly have good company in that category.) If asked for clarification, simply say, "I was terminated for violating X policy. It was a mistake that won't ever happen again." I know you are dying to share the story of how unfair it all was (and it was unfair), but it won't help you.
To avoid any surprises on the job hunt, call your former boss, explain that you are job hunting and ask what she will say if called for a reference. Note, this is not asking her to provide a reference, this is asking what she would say if called. A good recruiter will often track down a former manager even if she is not listed as a reference. If the reference is positive and accurate, great. Stop worrying. If it is negative and accurate, well tough. If it's negative and inaccurate then you've got problems.
You can mention that you obviously disagreed about the reason for termination, but that your performance was, overall, at a high level. Reminding her of the controversy over your termination will probably be enough to push her to knock off the negativity.
If you are worried about HR saying something, get a good friend to call up HR and state that she's doing a reference check and could they give her dates, salary, title and reason for termination. I doubt they'll do anything more than confirm what info she gives them.
(And as a side note, you only have 180 days to file a discrimination charge with the EEOC, unless you are in a state which has additional laws, then you have up to 300 days. This is something to keep in mind if this happens again.)