If we were uncertain whether he believes, perhaps we could leave him with his prior presumption of status. But if a person declares that he does not believe, only that he is uncertain—and doubt in faith is not faith—how can we leave him with his prior presumption of status?

He himself is unsure whether he believes. His fitness is determined by his presumed status, not by his belief.

Faith means that it is clear, or at least probable in my eyes, that God exists, etc. A person who is in doubt whether he believes means: I am in great doubt whether God exists—or in other words, I do not believe. In that case, it does not make sense to leave him with his prior presumption of status, does it?

For the sake of discussion, I will ignore the logical difficulty and the blatant tendentiousness in your words (a person who is in doubt is in doubt, not necessarily in “great doubt,” and still less has he already arrived at an atheistic conclusion, as you are putting in his mouth).
But beyond that, there is a difference between struggling on the way toward forming a position and a doubtful conclusion. When a person is thinking through his path, he does not yet have a position; he is on the way to one. It is possible that in the end he will remain in doubt, but inner conflict is not necessarily that. When he reaches a final position, it will be possible to determine how to regard him. So long as he is still on the way, there is a great deal of logic in leaving him with the halakhic presumption of status (as stated, the discussion here is halakhic, not about his artistry) that he had beforehand.