If he had answered "I'm a Methodist" or "I'm a Presbyterian" or "I'm a Baptist" I would have likewise rejected his application. He was asked a simple question "Are you a Christian". He did not give the correct answer. He gave a parochial answer to a broad non-denominational question. He should have known better. He didn't. He is not leadership material, especially for a group like the Salvation Army. Clueless people should not be in leadership positions.
I have no sympathy for Clark's predicament with the SA and his lawsuit is ridiculous. But not for the reasons you've given. If the SA don't want him because he is a Catholic that is just as acceptable as the Catholic Church refusing to employ someone who doesn't measure up to its standards or accept its principles.
I'm simply attempting to point out the somewhat nebulous meaning of the term "Christian" in common usage.