If one's flavor of Judaism incorporates a belief that is frowned upon by our misguided secular society and he is questioned on that matter, is he permitted, forbidden, or obligated to deny that he believes that?
Everything in the Torah has context and meaning. Often when something in Judaism is portrayed in a negative way it is a result of not really understanding what the Torah is saying. This is often as a result of something similar in another religion but enforced for a different reason.
For example in Islam the reason women must cover themselves up is b/c men can't control themselves and might be cause to rape them or think impure thoughts. At one point in America it was acceptable to think concern a woman who was sexually assaulted to say well look what she was wearing she was asking for it.
In Judaism the reason for modesty has nothing to do with the behavior of men who act badly but about the holiness associated with the behavior of being modest and having an internal connection to Gd. A big aspect of how women serve G-d according to Torah is from a deeper more internal way. This is unlike men who need to have outward signs such as a yarmulka (yire malka the fear of the king, tzitzis, tefillin, etc)
Rather than deny something in the Torah, there is an opportunity to correct a mistaken opinion causing a kiddush Hashem and "is honorable to himself and brings him honor from man" (P.A. ch 2 paragraph 1)