I think the problem is having something on your face everywhere you go. Simply making it smaller is not going to change that problem.
The important thing is to make the nose covering less obvious by making it transparent (probably as transparent as glass) and small.
But won't you have to put it on the same way as a face mask? Or do you have a different solution.
The nose covering won't need rope by the way... you just attach it to your nose and move on.
I can't imagine that type of face/nose mask. Do you have a sample illustration of how it looks like? the face masks that we are using for the protection of covid transmission are covering half of our face.
Transmission can be done through droplets which can pass through our nose and mouth. It is easier to reiterate the type of mask that you have in your thought if we can see it through an image.
Searched on the web for the exact model of what I described, didn't find anything close.
It can be somewhat shaped like human nose, covering from the top of a nose to the nostrils. It needs to be transparent like glass with filters at the nostrils parts.