does anyone make such a hat?

Not exactly within your parameters, but this eVent baseball cap was designed to work with the Spektr smock and might meet your needs: http://www.montane.co.uk/range/accessories/pace-cap

How about buying a breathable cap that you like and goving the bill a triple dose of DWR treatment? There are all kinds of cheap mesh trucker's caps. A mesh runner's cap or a sun hat like the OR Sun Runner could do double duty I fair weather.

Perhaps a silnylon sheath could be made for the bill, or even the bill and front panel?

I like idea. Many a bad hood has been fixed with a baseball cap. The bill would keep the spray off you face and glasses.

I have been eying this one for a while but have yet to buy one. Side vents and DWR and pretty light.

http://www.patagonia.com/ca/product/vented-longbill-cap-50-upf?p=29285-0-243

… ought to do the trick. I dunno what's in the bill but its waterproof. The outer fabric gets wet but you can angle it down so the water drips off. Havent noticed any wicking in either. Bought mine in '06 … I think the new ones are the same.

What if you took a cheap mesh trucker hat and coated the top of the bill with something like seam seal?

You could probably do the thinned silicone thing to coat the bill too. There must be some clear spray something or other that would do the same. All said and done, some heavy duty DWR like Tent and Gear Proof would be good 'nuff.

Most of these caps have a chunk of plastic in the bill. You just need to keep the fabric from wetting out and think about the seam between the bill and cap body.

I just remembered those painters caps at Home Depot. They are super light (and cheap) and have a plastic bill.

Not what you are looking for but REI sells an eEvent baseball cap. In a light rain or drizzle I find it more comfortable to wear my hood down I am cooler. When the rain gets worse I pull up the hood.