This conflict is originally introduced as a character vs society type, however, I think that as the story goes on and the theme recognized, we see more of a character vs self.
What is that?
It shook with his measured breath, as he gave out the psalm; it threw its obscurity between him and the holy page, as he read the Scriptures; and while he prayed, the veil lay heavily on his uplifted countenance.
I will not reveal my secret sins.
I will live the rest of my life with it in place!
Beloved and respected as you are, there may be whispers that you hide your face under the consciousness of secret sin. For the sake of your holy office do away this scandal.
No Mr.Clark you will not take my veil!
What crimes caused you to hide your face?
I believe that this is a satisfying ending for this story because it shows the 'ending' of the original black veil but also leaves something to be desired. Also, this shows the result of changing who you are, you are treated as an outcast, abandoned, and die alone, at least in this story line.