I believe, even as a non-Catholic, that the Pope of the Roman Catholic Church has the authority, if he chooses to do so, to resolve, at least for Catholics, hitherto disputed successions in monarchies that are or once were Catholic. He does not, however, have the authority to endorse the replacement of a Catholic Monarchy with a Republic, for such a development is intrinsically Evil and can never be approved under any circumstances.
Therefore, all non-sedevacantist Catholics must recognize Felipe VI as the legitimate King of Spain and Elizabeth II as the legitimate Queen of the United Kingdom and the Commonwealth Realms, since the Vatican does, regardless of what they think about the events of previous centuries. No dispute is justifiable. "Carlism" or "Jacobitism" that does not have the blessing of the Holy See has no reason to continue. But Catholics have absolutely no obligation whatsoever to obey Leo XIII and accept the French Republic or any other republic occupying the territory of what was once a Catholic Monarchy, just because the Vatican now also has diplomatic relations with those republics.