DIY question - dyeing a straw hat?

Okay LJ Hivemind, help me out here. I found a place that sells woven straw versions of my usual top hat. A straw version would be a very clever thing for me to have for whenever summer gets here and the weather turns unfriendly to me. However, the woven straw top hats are, well, straw colored. Which, yes, very steampunk-esque, but I'm not that steampunk, not to mention it would look odd with my black/b&w stripey/cupcake pink wardrobe.

So! I would need to dye it. Without damaging the shape. My first thought was spray-paint, but I suspect that would make the straw less breathable, and that's the whole point behind contemplating a straw top hat. Carefully painting it by hand with RIT mixed with alcohol? Carefully painting it with acrylic paint? I dunno. Which is why I'm turning to the hivemind for suggestions.

(alexiarnps, I'm especially looking at you for suggestions.)