If the Western bits of Roman empire truly were a separate empire then the vestiges of it had no power to give to anyone. The janitor of a corporation that closes down might be the last employee (because someone has to shut down the lights) but that doesn't mean that they are the new CEO who owns all the intellectual property etc.
The Catholic church only had whatever political power that was given to them by the state, the same as it is now in countries where it is the state church. IF the western Roman empire was a separate entity the authority ended when it collapsed. So even in your pseudohistory this doesn't really work the way you think it does.
Meanwhile in the real world, Catholic church, the way it exists now never held power in the Roman Empire. The Orthodox church has much stronger claim to that as they crowned the Emperors in Constantinople.