In America anyway, priests do not get special legal treatment over other religions. US Federal law does not care about canon law, and protects something known as “ecclesiastical privilege.” All private discussions between you and any clergyman of any faith are privileged information, inadmissible in court. We have similar privacy protections with other professions.
As regards Her Majesty’s government of Australia…So, if we are talking about a priest refusing to do something because of religious convictions…okay, fine.
Most of us are totally okay with people refusing to comply with legal requirements, even when there is a compelling state interest. Look at every pacifist religion. They refuse to do something because of religious conviction, and we let it go - even when every one else is mandated to do it. We have done so here since the American Revolution (which did have a few draftees, contra popular opinion).
If we are talking about people demanding to do something because their religion demands it, then that is slightly more complicated in US law, but clearly, it is a different issue.
Point being, if you are trying to make the argument that priests refusing to do something, will be analogous to Muslims demanding to do something or Satanists demanding to do something illegal (with clear and direct victims of that action), then I don’t think your fears are warranted.