I'm not religious but faith and calm can help terrifically to antidote fear and us bellowing at and around the pt.
I had the first and second editions of the orange book and do not remember baptism. I DO remember it from old first aid training. Maybe in orange book first edition (up to 1976 or 77).
To y'all, and again speaking as an unreligious man, how can it not be permissible to attempt extreme unction or baptism if the situation calls for it and all parties are, er, party to it? If the mother or parents of a fetus or newborn want it, what will happen if it is a little young, or a little out of wedlock, or whatever? What if someone gets faith on their deathbed and there's you and them, what's the harm, as long as you aren't proselytizing.(What do I know, maybe it backfires or something, but it's worth a chance).
I'll bet someone complains to your boss or the papers, especially if it is tax based care.
and the priest who was doing the deathn and dying classes was barred from the local ER's because he interfered with care.