It's probably also worth mentioning that a lot of the really big, zealous anti-witch movements happened a fair bit later than canon Ars Magica. Churchmen practiced magic of their own in 1220, and not just the divine variety.
The Church isn't canonically an enemy of the Order. They don't represent fundamentally opposed viewpoints. It's a bit like USA and China today. They aren't allies, but they have sufficient beneficial interactions that both have a lot to lose by conflict and a lot to gain by cooperation.
But when distilled down to a parish / covenant level - well, some get on and some don't. Hostilities are possible, individual churchmen agitating against magi is entirely possible. But a Crusade? Not without something pretty serious happening on both sides. I'd say there's about as much likelihood of a Crusade against the Order as there is of the Order and Church merging into a single entity - both are possible future events, but both would require a lot of work by entities who want to see that outcome.
I'm pretty certain there is a host of references (look at the cross-references) that say otherwise. The actual history of the Church got away with not being zealously anti-witchcraft in 1220 because magic doesn't actually exist; making it a reasonable thing for them to declare believing it exists a heresy on its own in real history. We don't have that luxury when the Order openly exists and isn't even vaguely secret about its true nature, an order of powerful & politically-independent (and inherently creepy) sorcerers that honour a pagan god. This is the time of the Schism, multiple Crusades, and deep-seated anti-semitism; this is not the atmosphere of an open and welcoming Church.
Obviously the status quo is that the Church isn't hostile to the Order, which says to me we need to come up with names/events/groups that explain this discrepancy.
Medieval people, also clergy, certainly believed that magic existed. For examples, have a look at the legends surrounding Gerbert d'Aurillac ( en.wikipedia.org/wiki/Pope_John_XXI ).
In real history, medieval bishops had to confront any kind of superstitions among their dioceses: also by declaring - with widely varying results - adherence to some heretical. In general they were pragmatic men, knowing both their duties and the limits of their authority.
We pretend many of these medieval legends and superstitions, not just anachronistic Hermetic magic, to be real in Mythic Europe. Do you imagine in Mythic Europe a Church in a permanent state of war with every hedgie, magical being or faerie?
The first thing you have to take into account is the historical, medieval Western Church itself. If you just picked up a few phrases from the Old Testament, interpreted them yourself, and then claimed the result to be the incontrovertible, valid instruction for the Church to treat a practical problem, that would have made you a heretic in no time.
Also there is debate as to what "witch" mans biblically. The word best translates to "whisperer", but was translated by Jewish scholars of the time into Greek as "phamika" (my spelling is probably off, even allowing for the fact it is the wrong alphabet) which I translatable into English as either poisoner or druggist. In any case it is clear from context it is talking about magic, but not magic in general but rather a single magical tradition, most likely the Chaldean practitioners of Babylon which was a political and sometimes military rival of Isreal. The word witch in English has a different meaning, but the word was translated as witch under some political pressure from king James when the King James version came out in English.
The word in the bible traditionally I understood to refer to a woman who has sexual congress with demons and/or sells her soul in exchange for magical powers. Of course, this understanding was itself post Mallus Malefactorium, which gain is a document which post dates Ar Magica and would later itself be declared heretical (primarily for stating that such women existed as well as giving a number of [strike]torture[/strike] tests to determine which women (and even men) were allie of the devil.
So exactly what a witch was in biblical terms for this point in time is very poorly defined, and probably not one of the most heated topics of debate. Sure the churchman nearest the covenant will probably wrestle with the meaning of the word if the covenant operate openly (and claiming the whole order operates openly is to ignore the vast amount of information in the books), but there is certainly not a church wide official position never mind a crusade.
Important note re: Catholic church - interpreting scripture in a way contrary to the official interpretation is the definition of heresy. Given that the Church hasn't declared a crusade on the Order in 1220, nor is it in open conflict despite being quite aware of the Order's existence, it's reasonable to assume that the Church's official stance is that hermetic magi are not, by default, required to be stoned to death.
This could well change in the course of a saga, but the change would require the papacy to choose to declare war on the Order. This isn't going to happen on a whim; parties interested in this outcome would have to work towards this goal because a lot of reasonable people from both the Order and the Church are going to realise that open conflict between the two groups is a Bad Idea.
The gathering storm-clouds of such a conflict and the PCs efforts to thwart / assist such efforts would make a perfectly legitimate saga.
That said, in your saga it's up to you how much the Church and Order don't get on. Being on the cusp of all-out war has a very 3rd-edition feel to it.
Historically, churchmen spent a lot of time trying to distinguish "natural magic", which was considered to be perfectly permissible, from the prohibited invocation of demons, with a quite wide variety of opinions occurring. Read, say, Kieckhefer's Magic in the Middle Ages (in the Cambridge Medieval textbooks series) or Valerie I. J. Flint's The Rise of Magic in Early Medieval Europe, for examples. Given that established historical fact, it's easy to argue that standard Hermetic magic is on the "natural" side. This is especially true when contrasted with the Order of Sueliman, whose magic almost perfectly fits the historical medieval Christian paradigm of demonic magic, since it utterly relies on summoned spirits.
So, the idea that Hermetic wizards are the true heirs of the Magi of the Book of Matthew, wielding natural magic with God-given talent in order to counterbalance to the diabolist malefici of the Muslim world, is not particularly implausible for 13th Century churchmen to come up with entirely on their own. And it's almost certain that various magi have made that point to churchmen themselves, so the idea is certainly going to be floating around. Given that, unless magi start interfering with the Church or preaching heresy, a consensus for a crusade isn't going to emerge.