That's a sound principle, as far as it goes. But it doesn't answer the question about a PARTICULAR war. If the bishop were correct that this is an unjust war, then he would be correct to instruct his people that they must not participate in it. In that case, "render unto Caesar" would be irrelevant. Caesar has no right to demand that the citizen do anything that is immoral.
As it is, the bishop is correct regarding the moral principles he discusses. He's just wrong about this war. It is not an unjust war of aggression.