I've always loved epiphytes (though I avoided orchids for years due to their difficult reputation). A few years back I bought a really small tillandsia bulbosa, used aluminum wire to hold it & made it so I could attach it to a brooch. It grew too large to wear within a very short time & I eventually lost it to bulb rot.
I agree it may be best to go with a straw-looking hat that's really made from woven plastic as that would last much longer. Another option would be to affix the tillandsia to a ribbon that can be removed from the hat for watering. And even attached to other hats or worn as a headband
I see no reason why you couldn't mount a micro mini orchid to a tiny piece of cork & wear that on a necklace or brooch or hat.
Also on etsy you can get necklaces made to hold tiny plants. Maybe you could use orchid media or hygrolon instead of soil if one of the little necklace pots seems suitable.