Would a spirit react differently to someone with the Power of Order than to someone without it?
I doubt that it would, since Power of Order isn’t required to exorcise (the whole question in Canon Law revolves around Power of Jurisdiction), and every pre-1965 exorcism formula contains within itself a conjuration.
Of course, though, we’re also looking at the fact a lot of the grimoires were written for clergy, at least one (Honorius) assuming Power of Order since it directs the operator to say Mass, while scholars such as Kieckheffer (Forbidden Rites, 1999) and Peters (The Magician, the Witch, and the Law, 1978) speak of clerical magicians — Kieckheffer claims clerical magicians were as scandalous then as child-molesting priests are now — we even see Chaucer’s parish-clerk Absalom resorts to a “true-love herb” (some texts say “talisman”, go to p. 70) to seduce the carpenter’s wife Alison (“The Miller’s Tale,” line 3692). Poem 54 of the Carmina Burana (“Omne genus demoniorum”) also takes the form of a magical rite, conjuring and banishing dryads and the elves Gordan, Ingordin, and Ingordan “by the seal of Solomon, by Pharaoh’s magicians,” and “through the miraculous and ineffable name of God, Tetragrammaton.” (my translation)
Finally, we have post-Reformation grimoires like the Key of Solomon frankly borrowing and adapting formulae from the Roman Ritual, particularly the exorcism of water before taking a ritual bath; the text is taken almost verbatim from the Ordo ad Faciendam Aquam Benedictam. This one’s likely a minor coincidence, but also an example that “official” Catholic ritual formulae still find a home in grimoire literature.
What I’m speaking to here is a documented existence (without implying tradition) of priest-magicians that was known and even ridiculed at the time, and we can see that many occultists desire to follow in those footsteps thanks to how many have been ordained in the Independent Sacramental Movement (a good introductory discussion is in Plummer’s Many Paths, 2005). In fact Gardner’s “Witches’ Cottage” was actually owned by an early ISM bishop, John S.M. Ward, before Gardner purchased it from him. If second fact, Gerald Gardner himself was ordained in an early ISM jurisdiction in 1946 — The Ancient British Church — a full seven years after he founded Wicca and six years before publishing Witchcraft Today!
The sheer number of occultists seeking Catholic or Catholic-ish ordinations tells me there’s a belief that a priest’s magic is stronger than a layperson’s. Even if the belief is unconscious or not widely held, these magicians wouldn’t be seeking Orders if the belief weren’t there.
Okay, what do I think? I think it’s a false belief. In fact I know it’s a false belief because the Power of Order is only considered necessary for absolution, transubstantiation, and perpetuation of ordination. Magic is not a sacrament but a sacramental, which derives from the Baptismal priesthood and not the Ministerial (this claim is sourced and documented in Chapter 3 of The Magic of Catholicism). In fact it bothers me that such ordinations are done in the belief of enhancing (or otherwise “adding to”) one’s magical repertoire and not in the context of actual pastoral ministry.
On the one hand, there’s a precedent. There is a tradition in the Western Church of ordaining monks to Holy Orders. These men are ordained without pastoral ministry and could be considered something of a template for what these magicians are seeking. In fact many ISM jurisdictions have a category referred to as “monastic ministry” wherein the priest has no congregation but offers his Masses for the benefit of the whole world in general and those who request his spiritual assistance in particular. What this means is that the framework is there.
While the framework is there, that framework is intended for the individual minister working toward sanctification within the Church’s own system in promoting the spiritual welfare of creation, and the only perceptible benefit is the ability to link his prayers with the Sacrifice of the Mass.
Such a man will (ideally) have been vetted over a period of several years before ordination, to ensure he has “the requisite knowledge and moral character” (1917 Code of Canon Law, 974.1; cf. 1983 Code, 1029). The idea is that a candidate, thus vetted, can be trusted not to abuse the Mass in reference to curses and black magic. This isn’t perfect (as many priests properly ordained have abused the Mass in this manner), but it’s better to have some system for screening out problem candidates, no matter how imperfect, than no system at all.

Abbé Guiborg, one of the best-known abuses of the Holy Mass
There’s also the fact that Christ didn’t give us the Sacrament of Holy Orders to “improve” our magical capabilities; in fact Jesus’ intention behind the Sacrament has nothing to do with any individual’s magical practice, period. He gave us the Sacrament so the Apostles and their successors (the bishops) would be full channels of his power as they were fruitful and multiplied his Church throughout the world. The willing transfer of the indelible Character of Order for the sole purpose of benefiting the candidate’s magical life is an abuse in my opinion, both because it’s based on a false belief (that priests are somehow “better” or “more spiritual” than laypeople) and subverts the intention for which the sacrament was instituted. Likewise, willingly passing on the Character of Order to a candidate who’s not fully vetted (a separate issue which happens in many ISM and even some Trad circles) is an even greater abuse.
Ultimately, though, the whole equation boils down to this: magic works on the principle of ex opere operantis, meaning a priest has never and will never have an inherent advantage over a layperson. If an individual priest shows greater talent than an individual layperson, it is either a case of the priest being born with “the gift” or of having worked his ass off for it. If a priest is lax in life and doesn’t work to develop his gifts, then a layperson who does will be a stronger magician. It’s that simple.
Yep, I’m sure somebody’s going to mention Leadbeater or some other clairvoyant in the attempt to refute what I just said. If you do that, you’re guilty of bad theology because you’re assuming private revelation has authoritative value in the public forum. I’ll stick with my almost 15 years of bearing the character of order on my own soul, alongside 28 years of magical study and practice. Coupled with the reality that I’ve yet to see any clerical magician perform better than a layperson without first having trained hard to get to that point. The weight of empirical data outweighs claims to private revelation every time.
My fellow magicians, don’t look for the “quick fix” when you’re just going to be disappointed anyway. There are no quick fixes.
My fellow bishops, you have full discretion over whom you’re willing to lay hands and for whatever reason you choose to do so. More importantly, those of you outside of Rome or the Four Patriarchates have the discretion to do so without fear of punishment from your superiors. I ask only that you bear in mind your awesome responsibility as gatekeepers and promote only those who are found ready and worthy of the irrevocable gift they stand to receive at your hands.