Actually, to the OP, I don't think that's a logical fallacy. It's a pretty straightforward argument:
-- You should stay away from people who are mentally diseased
-- apostates are mentally diseased
-- you should stay away from apostates
Nothing fallacious about it. Wrong, maybe, but each point follows. Now, an ad hominem would be something like:
-- apostates say JWs are wrong
-- apostates are mentally diseases
-- therefore, apostates are wrong
I think this would count, since even a mentally diseased person might make a good argument. But the claim that you should stay away from mentally diseased persons of which apostates are examples is not a fallacious argument.
1.The analogy is a false analogy. You can't "catch" mental problems, but, you can treat them. As shepherds who imitate The Fine Shepherd Jesus they are supposed to go after the "stray" and cradle them in their bosom to return the stray to the flock.
2.An Apostate disagrees and leaves. In the Christian context--why would a person who has dedicated their life to Jehovah see reasons to do so? The Watchtower article IGNORES reasons so that the issue of "just cause" won't be debated. Instead, they seek to dismiss without a hearing.
3.A physician doesn't stay away from afflicted people and neither did the Great Physician, Jesus Christ. He was criticized by Pharisees for eating and drinking with the WRONG SORT OF PEOPLE. If anything, the Watchtower article is an article those Pharisees would have written about Jesus.
4. The Watchtower's counsel goes against scripture.
Proverbs 18:13
New International Version (NIV)
13 To answer before listening—
that is folly and shame.
Jesus was considered to be an Apostate because his arguments with Pharisees and leaders of Judaism were condemnations they DID NOT WANT TO HEAR.
It simply never ocurred to Jewish Leaders that THEY COULD BE the mentally diseased ones. Neither does the Governing Body of Jehovah's Witnesses.