Considering Earth's growth beyond Religion (Or even disregarding that and considering all the Dominations we currently have) plus all the Religions of all the Star Fleet Member Worlds, I really don't see how a Position of Chaplain could be managed, you strip away the Denomination, and you simply end up with a Counselor, which we already have.
No, I think the only way to have a Religious person like that would be for them have a different function on the ship, say an Engineer who just happens to be Religious, and people seek out that Engineer because of their balance and their morals and ethics.
Or even perhaps someone with the Position of Counselor, who happens to be someone religious. Maybe even center some stories around Religious conflict for them. IE: Their Religious Beliefs lead them to believe the right thing to tell a patient is one thing, but, their Counselor Duties and teachings tell them they should be telling their patient the exact opposite.
So, yea, a Chaplain as a position, I don't think would work, you'd have to give them another position, but, they just so happen to be Religious.