It could be done so that equipping it removes any beard or facial hair, if any. How come other things like the nightingale cowl manage to remove the beard?
Do beards clip through closed face helms or the Nightingale Hood? If not, it seems that you could do the same stuff for the Shrouded Cowl as you would these kinds of items.
That's not a viable answer, as the Nightingale Hood is basically the same thing, and my beard doesn't clip through it's cowl...
Because closed face helms actually remove your entire head why do you think the muzzle on a beast race doesn't clip through? Those helmets actually remove your head. If the shrouded cowl completely obscured your face then like the nightingale hood it'd work fine, but it doesn't your actual eyes remain visible.
As I said before due to how beards were implemented the game engine can't simply remove them like it does the hair on the top of you're head. I've actually looked into the PC mods that "fix" this bug and they have beard clipping issues.