Hi John,
I too like cap-style brim hats b/c they mate better with other layers, but like you said, many are heavy cotton and not so breathable. I’ve taken to visor caps for this reason — my favorite is the one by Cloudveil (Snake River visor or something?). Nylon versions weigh about an ounce, and are much more versatile. Alone in warm weather, w/ a beanie and/or balaclava, even in the winter. It can be worn over thick layers, but since its only a front-brim hat, it easily fits under balaclavas and hats, too. Just my 2 cents.
-Jason