I too feel that it is impolite for gentlement to wear hats in class, and
have longed for the day when I would have the guts to say so in class.
Alas, that day has not yet come.  As it is, for years I was the only
professor in the college not to allow food or drink in the class, and was
considered "tough" because of it.  Just this year I relented, and now
allow drinks (but no food).  I have not yet decided on whether to go back
to the "no food or drinks" policy.
 
Here is a suggestion for the "hat" policy.... Put on your syllabus that
you suggest that students dress "appropriately" for class.  Also state
that you would prefer students not to wear caps.  Make no mention of it in
class .... and see what happens.  If anyone asks you about it, then give
your reasons.  Obviously students will see what you "prefer" (assuming
they read the syllabus), and may take action accordingly.  Or not.  Give
it a try.
By the way, I make all my students give a five minute speech, and the
instructions specifically say "dress neatly.  NO HATS!"  That is the only
time the "no hat" rule is specifically spelled out.
 
Good luck.  I would like to hear how it turns out.  JOHN NORWOOD