Should Catholics be concerned about the fate of the Temple Mount or the Al-Aqsa Mosque?

No. The temple should be of no concern to anybody, because it means nothing anymore. Why? Because Jesus is the temple, and the Church is the temple; the Jewish temple has been replaced. It is no longer functioning as a temple with sacrifices; there is no more Levitical priesthood, so they need no new temple. They are busy trying to find their red heifer so they can make a sacrifice, but that has no meaning whatsoever. It is just the last hurrah of the Jewish people, because now that they have their nation, they want to show off to the world what they had in the past and think it is something for the present, and it is not. I would not be surprised at all if they eventually replace the Muslim mosque with a new temple, but it would still be their last hurrah, and they would make the best of it as they can.