Yes, the designer in this project naturally wanted both a border and a box shadow. I don't know how to accomplish that in CSS without using additional elements. I'm currently using strategy #1, hoping that nobody notices :)
As in the elsewhere linked CSS Hexagon tutorial (see last comment on http://jtauber.github.com/articles/css-hexagon.html) it's noted that there's a hexagon character. Well there are also triangle characters - which you could apply a [or multiple] shadow[s] to.