I'm not familiar with that verb in the context of happening to one's neck. Would my suggested design be different, in that context, from existing helmets that have a stiff aero protrusion in the rear and a stiff brim in front? Also know that the full-brim could be made out of the same crushable foam as the helmet itself.
As for DaBrim, it's TOO big. It's overkill for shading one's face.
The full-brim of the hard hat is just enough shade, considering that the bicycle helmet is already about an inch thick. A brim of 2.5" was typical for a fedora, back in the heyday of mens hats.