stiff brim I can’t offer any ideas about a neck drape, but if you want a stiff brim, and a great rain hat, consider a Filson in tin cloth. My #315 has a 3" brim and would probably stand up to that much wind (I’ve never tried to measure it.)
If its raining hard, and I’m outside, my Tilley stays inside, I’m wearing the Filson.
It is too warm for just a sun hat. Tilley’s are better for that.
Grow hair long or Patagonia Bimini. you can always push/feed a piece of line or wire through seam of the Foreign Legion part to make its perimeter a little heavier so it does not blow around.
Best / ugliest hat in the world? This hat is butt-ugly, but I think it will do what you want, but did I mention it’s butt-ugly? The inventor has given it a name “The Best Hat in the World”, but geez, did I mention that it’s ugly? YMMV…