A small brim would be good. I have a "Radar" cap I use in the winter. It was meant to be worn under the M1 helmet of WWII. It was made famous by Radar from MASH. The one I use I got from a WWII reenactor supplier, 100% wool. When I paired it with my Asbell Jacket it kept the hood from falling down into my face keeping my vision clear and added a bit more warmth to the set.
Calling it the Asbell beanie would be nice, simple, and marketable. If you could add a short brim then you could call it the Asbell short-cap.