I like the Sitka Gear Jetstream hat. It's a very short-brimmed hat that's soft and very comfortable. It's warm and the short brim would not interfere with a bowstring and if it did, it would be so soft as to bend out of the way. It can also be bent up. It is super light. The Jetstream uses Windstopper Soft Shell so it's more water-repellent than waterproof but an application or two of waterproof spray should boost it to the level you like. Personally I use it over a Merino or other Sitka Beanie or the Traverse Balaclava. When it's really cold, I put an older Celsius hat (or a hood) over the top of it too.
Herd Bull (Mike Mitten) is a big believer in wearing numerous layers on our noggin, and I've noticed by doing so that the rest of my body seems to stay warmer longer as well.